Basically players team up (usually Federation vs Klingon though sometimes one Klingon house vs another) and fight each other. For a PvP match to be played you need a certain number of players on each side. There's a PvP Queue option in the list that pops down when you hit the arrow at the bottom right of the mini-map. Pick a map and it drops you into the queue - waiting until enough other players sign up.
Hope this helps ... and hope you've been enjoying the game.
What is the PVP and how do you play levels with other players, or develop other types of friend live play?
The game has options to invite people to join your team, or request to join a team. There's also a search function in the game to find other players. Once teamed up, you'll see that the game also has the ability to artifically boost or lower level to that of your team leader - which allows players to do their friends' missions, and so on.
That's the mechanical side of things.
But how do you actually go about teaming up with people? Well, you can ask - that's one use for zone chat. Go on there and ask other players if anyone's interested in teaming up. It's like any other social interaction, really; you gotta sell yourself a little.
Stromgold is right. Joining a Fleet will help you a lot. A good Fleet with good people should be very understanding in showing you the ropes.
I just got promoted to lt 11, and a new ship, but the only thing i seem to do is fly around getting missions and pressing the fire button in space and on the ground. !
Welcome to the world of MMOs - the basic drive in the majority of MMOs is to level up by doing missions and other things that advance your characters abilities (like crafting, etc.); and get you 'gear' to better outfit yourself, your ship, and your BOffs (Bridge Officers). Oftentimes as you've probably already noticed, it's not as straightforward as a single player RPG as it tries to make it more opened ended for you as far as what you want to do (ie the 'order' you want to take and do missions, etc.)
As others have said, look to join a good Fleet, and look for fan internet sites that have info on the game (Yu should be able to find sites discussing skill build strategies, etc.; that discuss what the abilities do so you can make a better decision on what type of satuff fits you playstyle.)