I quite recommend using Nexerelin. That mod quite increases your and existing faction's option in the base game. Makes it a bit more complex, but also gives you the ability to deal better with the situations. For faction mods I like Free Stars Union and the Mayasuran Navy mods the best. Those factions do not radically different to vanilla factions, and I think they have overall a decent balance with their ships and weapons. A bit greater change would be the Interstellar Imperium mod. I quite like the weapons it offers, but it also offers a bit more non-standard playstyle compared to vanilla. Also, you've got the Ships/Weapons pack, which does just that, adds extra ships and weapons to vanilla factions.
For other mods I'd reccommend Galactic Markets, which basically gives you an easily accessible list of all commodities, making trade plans easier and more thorough. Common Radar and Combat Chatter are nice additions as well. Radar gives you just that, a small radar you can see during combat as well, and Chatter is a decent immersive mod, making your ships comment on the battle and the current situations they find themselves in. I also use the Interesting Portraits mod, which gives you just that, more portraits to be used by you and other factions. These mods do not really influence basic gameplay, they just add a little bit of extra.
Also check this sticky: https://fractalsoftworks.com/forum/index.php?topic=177.0
Also, as quoted above, there's a rather useful Mod Index. Browse around there, and pick whatever sounds more fun.
Oh, and you'll have to add some non-standard mods as well, so your first step should be to download LazyLib, MagicLib and GraphicsLib. My rather simplistic definition for them would be, that they provide toolsets, commands and functions other mods use. And if you've got weaker hardware, like me, you'll likely have to do some fine-tuning for GraphicsLib. How you can and should do that, you may find here:
http://fractalsoftworks.com/forum/index.php?topic=10931.0