I'd play through PvE and learn the basic machanics of the game and unlock a few skills before starting PvP. If you do start PvP and want to do high-level PvP, you'll really need most skills unlocked and all three chapters.
It is possible to have a PvE character PvP, buut I highly reccommend you do not. PvP characters are so much easier to PvP with and have many more things unlocked than a PvE character would. Just leave one character slot open for a PvP character that you can delete and remake whenever you want it to switch professions.
I agree with Lotrfish. It is better to learn how the game works and unlocking skills before going PvP, but if you`re a dedicated PvP player, you can buy PvP Edition games through the online store, they let you unlock every skill, and your Heroes get to use them also, IF you have Nightfall.
I recommend PvE because it allows you to play through all the game. PvP is just a limited (but important) part of the Guild Wars universe. When you leveled to 20 in PvE and got ascended, you can also get all the PvP goodness with the same character.
Also, Lotrfish's statements are somewhat inacurate. A PvE character can easily PvP when you are new to the game doing more novice PvP type areas (Alliance Battles, Random Arenas, and unranked Hero Battles). It might take a while for your PvE character to be used for high end PvP, but feel free to play around with PvP at any time with your character