PvP works by joining Queues (it's on the same menu set as the C-Store, the V triangle in the mini-map view). Join a match, earn PvP badges. Each map is your basic "kill the other side" with either a capture the flag or a simple timer goal (at least those that i've joined).
PvP earns you PvP badges which you can then exchange for usually better gear than what you can get from exploration badges.
As to the ship question, since you can't fly a ship higher than your rank, just stick with the one you are given for free at each promotion. The only time you'll need credits to buy a ship is if you felt you chose the wrong one for the freebie (i.e. got a cruiser then felt you should have gotten a science vessel).