Well, we have quite a few elite STF runs (mostly space) and want to also venture more into ground elites. To be fair, teh Fleet actually takes part in most things the game offers, but we do have some good team runs on STFs.
We don't focus solely on STFs, but you should still check out the United Federation of planets (UFPlanets.com). We have players in most timezones from the casual to the hardcore, so you can usually find someone else online to group with; We actually have three in game fleets: a fleet each for main characters and alts, and a klingon fleet, but we have a shared chat channel to keep in contact with each other as if it were a single fleet.
We have weekly STF events on Wednesdays and Sundays sponsored by our STF unit, but there is usually a STF group running STFs at any given time, and if there isn't, all you have to do is ask in chat and you can easily throw a team together.
The members are friendly and helpful and will gladly share any advice if you need it, and we run a Teamspeak server, though it isn't mandatory to use it for most things, and microphones are never mandatory.