We play a version of Star Trek called Spheres of Influence, but it is home-made, so that won't do you much good. I mention it though because I basically agree with 1of3, nearly any good system will do ya depending on how you want to adapt your basic setting.
For instance do you want the game to take place in Federation Space primarily, along the frontiers, or both inside and outside of Federation space, like our game does?
Do you want it to deal typically with standard model starships and basic internal Star Fleet concerns (politics, border patrol, basic exploration, warfare, inter-species relations, societal issues - of any era) or do you want it to deal mostly with experimental technologies, covert contacts with aliens, advanced alien technologies, proto-type experimental star craft, and exploration deep outside known space? Or do you want it to be a mixture of aims? (Actually our setting is more a mix or hybrid between Star Trek and Traveler.)
Depending on your intentions, and what your players want, it is often best to create your own setting using the Federation as a baseline, or control setting, upon which you create your own milieu, or to which you make your own particular setting adaptations.
Either way, I can suggest this without hesitation. When it comes to tactical ship combat, or hostile ship to ship encounters, then sue Star Fleet Battles (adapted to your setting and era) because that is a far better naval combat system than any RPG version I've ever seen. It's easy, flexible, intuitive, fast, and you can quickly and fluidly add in things like Advanced ECM, stealth functions, change ship designs, add or subtract experimental weapon systems, vary plans, create new alien ship models, etc, and you can do it all on graph paper without much effort. The combat system makes it easy to track combat as well, and the whole affair can be easily seen and visualized.
We role play Spheres of Influence, but when it comes to ship battles, we play Star Fleet Battles.
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