Let me start my answer by saying that Irish mythology is a complete mess, not in the sense of the myths per se, but of the sources. The division in the four major cycles is a convenient cataloguing tool, but the contents of each cycle are far from being coherent and homogeneous stories that can be easily put together in a collection.
To give you an idea of how vast and fragmented the corpus of sources is, here are two websites which give a very comprehensive list of both original and translated medieval Irish texts:
Both websites offer very reliable translations of stories from all the four cycles (the second one in particular is from the University College of Cork, if you are worried about the reliability of its contents).
My recommendation, regarding the Ulster cycle, is to start with what is considered its central text: the Táin Bó Cúailnge. The translation by Thomas Kinsella (titled simply "The Táin") is very good and easy to find. Then, I would read "Early Irish Myths and Sagas" by Jeffrey Gantz. It's a collection of translated stories from the Ulster cycle with extensive commentary, which is still used in universities as an introduction to the cycle. This should cover the most popular stories from the Ulster cycle: you can then explore the lesser-known ones using the websites I linked above.
Regarding the Mythological cycle, the two main texts I recommend reading would be the Lebor Gabála Érenn (also known as the Book of Invasions), whose translation by R.A.S. Macalister you can find in the first website I linked above, and the Cath Maige Tuired (or the Second Battle of Mag Tuired), whose translation by Elizabeth Gray is in the second website. Again, you can use both websites to find more minor texts from the Mythological cycle.
I cannot recommend anything regarding the two remaning cycles, as my knowledge of them is very limited. However, I would like to mention that there is a collection of translations of tales from all four cycles by Cross and Slover called "Ancient Irish Tales", which I have seen receive a lot of praise both for the quality of the translations and for its comprehensiveness. I haven't read it yet, but I plan to in the near future.