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.