Project Navras may very well be the fleet you're looking for. We're a fleet nearing 70 ships, at the moment, devoting ourselves to enjoying every aspect that Star Trek Online has to offer, whether it's RP, PVP, or simply good, old fashioned PVE, with raids and fleet actions. We have our own forums, voice comms, and weekly schedule of events-- with member access to the Fleet Bank, and regularly-held contests for our members.
We have a complete award/medals system on our forums, thus allowing our members to earn commendations and decorations from their actions within the fleet. We also have divisions, allowing you to devote yourself to certain duties/responsibilities within the fleet that will help you in progressing through our ranking system.
If you're interested, then it all starts by checking out our fleet recruitment thread. The link is located in my signature-- and after that, check out our website and forums at www.ProjectNavras.com -- you might even decide to fill out an application.
If you have not already chosen a fleet, I ask that you please consider SinistaR fleet. We are a casual fleet who focus on quality than quantity in our members. We pride ourselves in offering a fun, helpful and drama free fleet where members can hang out, run raids, and enjoy each other company.
If you are interested, please send me a PM and we can set up an interview so that we both can better get to know each other.