The STFs are a must. They're very challenging and lots of fun.
Play lots of PvP, and talk to as many people as you can to try to improve your game. Kerrat is the best as you dont have to queue. Just dont let all the trash talk get to you.
Grind Starbase 24 for dem purple lewts! Or just to make enough credits to buy stuff from the exchange.
Fly into low-level DSEs and just heal newbies.
Or, yeah.. make a Klingon. The sorties are awesome.
Edit: Oh, and visit all the open areas, if you haven't already. I didn't go to Vulcan until after I made VA. The little diplomacy mission there is cool, and all the NPCs have little tidbits of Star Trek lore to talk about. Kinda cool.