We have several details connected with the part of the chronology in question, but as far as I've been able to find, beyond plain statements about there having been a cosmic conflict, there are only two fairly fleshed-out fight scenes from this war. One of them is the final showdown between the Olympians and the Titans, narrated at some length in Hesiod's Theogony 664-735.
Kronos vs Zeus
The other battle-scene is a brief mention by Nonnus in Book 18 of his Dionysiaca. Kronos is here described as having the power to control weather phenomena so that in Lines 230-234, he calls sharp arrows of hail down from the sky upon Zeus but then the latter, in response, "armed himself with more fires than the sun and melted the petrified water with hotter sparks." I presume that Zeus's "fires" here are intended as a poetic description of lightning. In Dionysiaca 24, Kronos is "armed in vain with the watery weapons of the storm." In Book 12 of the same work it is implied that among these weapons are snowstorms.
Nine Hundred Arms
Apart from Poseidon's Trident, Hades's cap of invisibility, and an unnamed giant connected with Kronos (more on him in the section
Kronos's Gigantic Allies below), the only other thing we are told about the implements of warfare employed during this altercation is that both sides used hard bélea, "missiles," of some kind, against each other. Occurring in this instance in Hesiod's Theogony 684, this word often refers to arrows or spears but could mean any kind of weapon, such as swords, or even scorpion-stings. In Line 716 the term refers to the barrage of rocks hurled at the Titans by the Hundred-Handed Ones.
Hesiod's narration of this battlefield clash is concerned mostly with the environmental effects thereof. According to this passage, the two rival groups caused so much commotion that its reverberations were felt throughout the universe. The noise they made rang throughout the sea, the sky quaked, the stamping of their feet shook the earth all the way into Tartaros [Tartarus], and the din of their voices reached the stars.
The first few lines of the aforementioned Theogony passage tell us that Zeus's entire team of gods, "both male and female," participated in the battle, but the only figures who feature in their own special cameos in the ensuing combat as to be singled out by name are Zeus and his three Hundred-Handed uncles. (See, however, the case of Menoitios [Menoetius] below.)
After the world-rattling noise made by the clashing armies, "Then," in Lines 687-714, "Zeus no longer held back his might", whereupon he began to pour a blitz of thunder and lightning upon the enemy. Even so, Hesiod's focus remains fixed on the damage wreaked upon the landscape, which was bathed in flames that lit all the woodlands ablaze.
Meanwhile the entire atmosphere was cooked all the way up to the limits of Chaos itself so that the sea and even the perpetually placid, earth-encircling stream of Okeanos [Oceanus] begin to boil, and the Titans were blinded. In spite of all of that, and further noise, dust-storm and earthquake, the contest raged on as the two sides "kept at one another and fought continually in cruel war." Kronos did not surrender.
According to Theogony 713-735, what turned the tide of the war is that Kottos [Cottus], Briareos [Briareus] and Gyes, the three Hundred-Handed Giants, overpowered the Titans with the superior speed and volume granted to their firepower by the number of appendages they had at their disposal for launching their "missiles" of stone.
The scene then abruptly shifts to a description of Tartaros and how the Titans were cast into it after being assaulted by the Hekatonkheires [Hundred-Handers]. Poseidon appears in this passage, but only to be mentioned as the builder of the bronze doors fitted over the mouth of the pit of Tartaros to increase its inescapability. The passage then ends with the Hekatonkheires being sent back to Tartaros but this time as its jailers, guarding their brother Kronos who previously had held them prisoner in the same chasm.
There were four of Zeus's cousins who were brothers, named Atlas, Menoitios, Prometheus and Epimetheus. In some way or another, whether they had sided with him in the war or not, Zeus ended up having a major falling-out with each of them that resulted in some punishment or calamity that the Olympian leader visited upon each brother after the war.
Of the four, Menoitios is the one we know the least about. He is, however, exceptional in that he is also the only Titan specifically named as having been brought low by Zeus's thunderbolt during the course of the war.
Menoitios was hubristic, and far-seeing Zeus struck him with a lurid
thunderbolt and sent him down to Erebos [Erebus, the Darkness of the Underworld] because of his mad presumption
and exceeding pride.
... Menoitios, he whom Zeus in the battle with the Titans smote with a
thunderbolt and hurled down to Tartaros.
Apollodorus, Library 1.2.3
For some reason William Smith's 1890 Dictionary of Greek and Roman Biography and Mythology goes so far as to say that this lightning-zap actually killed Menoitios.
Sykeus, the 13th Titan?
Athenaeus' Deipnosophistae, quoting Tryphon's History of Plants, quoting The Farmer's Handbook by Androtion, refers to Sykeus as "one of the Titans" whose mother was Gaia (the Earth), essentially suggesting the idea that there was a thirteenth Titan born to Ouranos [Uranus, the "Sky"] and Gaia, although Ouranos is not explicitly mentioned in this passage. We know nothing about what happened to Sykeus beyond the assertion that Zeus pursued him and that Gaia protected him, causing the sykea ("fig-tree"), "to grow for her son's pleasure; from him also the city of Sykea in Cilicia got its name."
The dawn of war is described in Theogony 383-403, where we learn that the first soldiers to join up with Zeus, upon his declaration that he was going up against his father, were, incidentally, a bunch of winged Titans. These recruits, two sisters (Nike and Bia, "Victory" and "Violence") and two brothers (Zelos and Kratos, "Zeal" and "Might"), were the children of the Underworld river Styx, and eventually would become the personal bodyguard of Zeus.
There's no account of any confrontation between Poseidon and Okeanos because the latter abstained from fighting in the war. He was also the only one of the original male Titans to refrain from laying violent hands upon Ouranos when Kronos was castrating their father.
When Okeanos learned of Zeus's war announcement, he advised Styx, his eldest daughter, to enlist with the Olympians along with her four children. As a reward, Styx's name became the most sacred oath in the universe, and, once Kronos had been overthrown, Styx's offspring were assigned their permanent employment at the side of the new king of the gods. Okeanos, for his part, maintained his position as the god of the world-encircling river which was his own body, and thus an integral part of the superstructure of the universe.
Zeus had incentivised his soldiers with the lure of positions of privilege in the new régime for whoever would join his cause. According to Aeschylus' play Prometheus Bound, Zeus promptly kept these campaign promises by doling out the aforementioned goodies "as soon as he had seated himself on his father’s throne".
In a fragment from a lost work of Aeschylus, the goddess Dike, "Justice," says that Kronos was defeated by Zeus because the latter had her on his side,
"since his father began the quarrel, that he paid him back with Justice
on his side. That is why Zeus has done me great honour, because after
being attacked he paid him back, not unjustly."
For this reason, says Dike, she sits "in glory by the throne of Zeus".
The battlefield of the Titanomachy would appear to have been the Plain of Thessaly, which lies in between the two mountains upon which Hesiod situates the strongholds of the two opposing camps. The Titans were stationed on Mt Othrys, at the southern end of the plain, while Zeus and his crew had set up on Mt Olympos [Olympus], towards the northern edge of the region.
In the same passage in which Hesiod supplies us this geography (Theogony 616-643), he says that the fighting between the two camps never let up, not even for a little bit, but raged on "ten full years ... continually", with neither side losing or gaining any ground because of how evenly they were matched. It was then that Gaia advised Zeus to release the Hekatonkheires from Tartaros, in which incarceration they had spent their entire lives, and where Kronos had posted an immense millipede-like monster called Kampe [Campe] as their prison warden.
According to Apollodorus' Library 1.2.1, Zeus killed Kampe in order to effect his uncles' release. In Dionysiaca 18.235-267, Nonnus describes the scene at length. He does, however spend the majority of those 32 lines merely describing the monster in intricate detail. Towards the beginning of the passage he spares a few words to say that Zeus "destroyed high-headed Kampe with a thunderbolt". Thereafter the passage concludes with the very scant statement that, simply, the god killed and conquered her.
Apollodorus has yet way less detail than this, although it is from him that we learn why Zeus had needed to confront her in the first place. In Apollodorus, the Cyclopes as well as the Hekatonkheires have likewise endured lifelong captivity in Tartaros before being released by Zeus. At this point Hesiod's narrative continues with the Hundred-Handers being fed nectar and ambrosia, which "revived their proud spirit within them".
After this it doesn't take much for Zeus's two sets of triplet Giant uncles to get with his program, and this is where Kronos's team virtually gets pelted into Tartaros after being sprayed with thunderbolts.
In Hyginus' Astronomica 2.39 the Cyclopes construct the constellation Ara, the "Altar," upon which
the gods are thought to have first made offerings and formed an
alliance when they were about to oppose the Titans... From this
observance men established the custom that when they plan to do
something, they make sacrifices before beginning the undertaking.
According to Prometheus Bound, Prometheus learned the future from his mother Themis, and together with her he defected to the Olympian side. Before he did so, however, he tried to persuade his fellow Titans to do likewise but they wouldn't listen to him. Hesychius says that Prometheus had been the herald of the Titans.
(There is an obscure apocryphal story according to which the reason that Zeus is hostile towards Prometheus is that he was born to Hera after the goddess had been raped by the Giant Eurymedon while she was still living in her parents' house. Eurymedon was cast into Tartaros together with the Titans who lost the war.)
There was also at least one defection to the other side. According to Ptolemy Hephaestion, the winged rainbow-goddess Iris, who was the swiftest being in the universe, had a sister named Arke [Arce]. Like Iris, Arke was winged, and
during the struggle of the gods against the Titans, Arke flew out of
the camp of the gods and joined the Titans.
After the war, Zeus tore off Arke's wings and cast her into Tartaros. Thousands of years later, at the wedding of Peleus and Thetis, he presented these wings as a gift to Thetis. When the sea-goddess had her son Akhilleus [Achilles], she attached the wings to his feet, thus making him the fastest runner alive at the time.
Kronos's Gigantic Allies
Like Zeus, Kronos too had Giants on his side. In Dionysiaca 18, immediately after the reference to Kampe's death, Nonnus tells us that Indos, the ancestor of the Indians, was an autochthonous "champion of warfare with towering limbs... brought low" by Zeus "in the conflict with Kronos".
Son of Ekhidna [Echidna]
He then goes on to describe a humongous "dragon-footed" creature he calls simply "Ekhidna's son" and "the enemy of the gods", who spat venom into the Sky in such profusion that it created large clouds of poison, and who was so huge that the birds flying through the air would get caught in the vast mass of his dreadlocked hair, which hair he would then use to sweep these birds into his mouth to chew them live for dinner.
Nonnus seems to be describing a peculiar apparatus in which Kronos used this Giant as some sort of missile or shield to counter Zeus' thunderbolts. The Giant was killed by Ares, and as is typical, we are told nothing about the method by which this was accomplished.
Briareos-Aigaion, the Sea-Giant
There apparently was a rendition of this whole story in which a different version of the Hekatonkheiros [Hundred-Hander] Briareos took Kronos's side in the war. We are told in Iliad 1 that Briareos is this Giant's divine name but that "all men" know him, rather, by the name Aigaion [Aegaeon].
There is a now-lost Titanomachy epic attributed to Eumelus of Corinth. One of the very few things we know about it is that, within it, Aigaion, just as in the Iliad, is depicted as dwelling in the sea, but unlike in Homer's work, Eumelus says that Aigaion was "an ally of the Titans." He also portrays the Giant as a son of Pontos [Pontus, the "Sea"] and Gaia. According to Ion of Chios, Aigaion was the son, rather, of Thalassa, the female personification of the sea (over against the male Pontos).
Callimachus' Hymn to Delos seems to envision Briareos as having been buried under Mt Aitna [Aetna] on the island of Sicily, in the same manner that the same end is usually ascribed to the Giants Enkelados [Enceladus] and Typhoeus after their altercation(s) with the Olympians. Earthquakes and volcanic eruptions on the island are then explained as one of these monsters shifting under the earth's weight above them, or fuming furiously against Zeus.
The Guts of the Serpent-Bull
The featured contribution of this alternate Briareos to the war effort during the Titanomachy is narrated by the Roman poet Ovid in Fasti 3.793-808, where he says that Gaia had given birth to a shocking monster in the shape of a bull that, in place of hindquarters, was composed of a snake. It was somehow known that whoever burned the intestines of this creature "was destined to defeat the eternal gods."
Styx, trusty ally of Zeus that she was, and aided by the counsel of the three Fates, managed to capture and corral this monster snake-bull in a location which seems to be the Underworld. Briareos, however, was able to kill the monster using an unbreakable axe. He was about to perform the dread sacrifice, but since Zeus had commanded the birds to bring the monster's entrails to him, the kite snatched them just in time, delivering these innards to Zeus. This bird was then commemorated by being placed in the sky, within the constellation Boötes, as an asterism named Milvus, the "Kite."
A fragment from an unnamed writer says that a Giant named Azeus son of Gaia "grew to manhood amid the mighty battles of the Titans." The most natural reading of this is perhaps that Azeus just happened to turn 18 or 21 while the Titanomachy was raging and not necessarily that he participated in it at all. And if he did fight in the war, we do not know on which side.
According to Giovanni Boccaccio's Genealogy of the Gods of the Gentiles 3.5, Akheron [Acheron], the Underworld River of Woe, was a son of Demeter "without a father," or he was the son of Helios and Gaia, and originally lived in the upper world. During the Titanomachy he provided "clear water to the thirsty" Titans. After the war Zeus muddied his clear water and thrust him into the Underworld where the sole source of moisture for his stream now became the tears of sinners suffering in Tartaros.
The God That Fought for His Life (and Lost)
There are yet other, rather different accounts of the conflict between the new gods and the Titans. At least two of them might be Orphic. In Dionysiaca 6, Nonnus tells the story of Zagreus, the powerful shape-shifting baby son of Zeus by his own daughter Kore (who later becomes Persephone). Indignant that Zeus has placed this illegitimate infant upstart upon his own throne and lets him play with his thunderbolts, Hera induces "the Titans" to trick the baby god and cut him to pieces with "an infernal knife."
The Titans are able to corner Zagreus by covering their faces
with chalk and by showing him a mirror...
Zagreus's first instinct is to transform his appearance into that
of his own father Zeus... Then he appears as... "Kronos, heavy-kneed,
and pouring rain." After those... disguises, Zagreus becomes a
baby, an adolescent boy, a roaring lion, a wild horse, a snake, a
tiger and then a bull.
With the sharp horns of his bull-form, Zagreus attempts to gore
the Titans, but then Hera, who is watching from a distance, emits a
bellow so loud that it causes the Zagreus-bull to collapse. The bull
then gets slaughtered.
~ From my Answer to another
In response to this, Zeus confines the Titans to Tartaros, and angrily sets the entire world on fire until Okeanos weeps for the scorched earth. Zeus calms down and then, in order to douse the flames, he floods the entire cosmos, causing even more death and destruction than the fire did. In this version, this is how the Flood of Deukalion [Deucalion] happens, after which the land is drained of the floodwaters, and the world gets slowly repopulated. Zagreus himself is later reincarnated as Dionysos [Dionysus].
In a variant of this myth, the Titans devoured the murdered Zagreus-Dionysos, and Zeus reduced them to ashes with a blast of lightning. These remains were then mixed with mud, from which arose humankind, which therefore has two warring natures within it, one celestial and Dionysiac, and the other earthly and Titanic.
The African Titan-Wars
According to entry 150 of Hyginus' Fabulae, Hera was jealous of Epaphos [Epaphus], yet another one of Zeus's illegitimate offspring, in his case because he ruled the great kingdom of Egypt. Hera assassinated him and then encouraged the Titans, led by Atlas, to depose Zeus in order to restore Kronos to the throne. Attempting to "climb up to heaven," the Titans were instead cast headlong into Tartaros by Zeus, who was supported on this occasion by Athena, Apollo and Artemis. Posted in what is now Morocco, Atlas was then forced to bear the weight of the sky on his shoulders in order to keep it separate from the earth.
In the Library of History 3.68-74, Diodorus Siculus narrates a euhemerised version of the Titanomachy in which Rhea was first married to a Libyan king called Ammon. After an affair with "a maiden of unusual beauty whose name was Amaltheia", Ammon had a son named Dionysos, whom he hid from his wife. When Rhea found out about it, she went to her brothers the Titans and married one of them, Kronos.
When Dionysos had grown up, war broke out between him and his father's ex-wife, who, together with Kronos and the rest of her brothers, began an uprising against the rule of Dionysos. All the characters in the story are presented as regular humans. At some point Dionysos' army is composed of both Libyans and Amazons. He overcomes the Titans, making peace with Rhea but eventually having to kill rebels led by Kronos.
Parts of the above have been adapted from my participation in a post about the Titanomachy on MythForum.com