Swtor has a few diffrent types of quest chains and all of them are optional except the basic game tutorial (levels 1-10, along with your starting planet quests.) Once you breach the level ten mark, you can choose which class direction you will be taking and the entire game is open to you at that point. (Actually, if you can convince somebody with a higher level avatar to kill everthing and help you run through your class stuff, you could make it up to the fleet and into a pvp match as early as level 5, although I am going to have to make an alt to confirm that.)
You do not need to quest to get your ship as long as you are willing to bum rides on other people, although if you are devoted to no questing, I can't see why you would possibly need to get to any of the other planets. The Imperial fleet has everything a fine solo pvper like yourself could possibly need. You make more than enough per pvp match to repair your gear and you have the GTN to buy anything you might need that you can't obtain from the pvp gear vendors.
If you can stomach questing until you get your first and second companions, you can go down to the crew skill trainers and pick up two of the Mission based gather skills and send your companions out on those to generate even more supplemental income for all that gear you are going to have to buy since you aren't going to get the normal free gear that people get as quest rewards. The good news is that pvp gear is the equal of the other high level gear in the game, so if you decide to something else at fifty you will be able to.
Additionally, at a mininium of ten grand experience per ten minutes, pvp is one of the fastest way to chug down experience, so you should be hitting fifty faster than any other way (looking at my experience/time with my questing alt, which is mostly solo, with few flash points thrown in and some crafting to get to 400, I am getting about 3000 experience every ten minutes while actively questing.)
I will update this with more firm facts & figures once I run to fifty doing the pvp only.