Play one PvE character for 20-30 hours. That should give you enough skills to do ok in Competition Arena (make sure to spend ALL your skill points before you delete a PvE character). From then on, you can make PvP characters and use them to gain faction to buy new skills (hint: if you choose one of the premade characters, you can use the skills they have even if you don't play the same build).
Here's a list of quests that give skills:
http://www.guildwiki.org/index.php/Skill_Quests
Getting as many skills as you can through quests will save skill points. Generally, skill trainers in the main cities will have skills that you can get through quests, while skill trainers in minor towns will have skills that you can get no place else in PvE.
Here's a list of skill trainers, so you can spend your skill points more efficiently:
http://www.guildwiki.org/wiki/Trainer_locations
Also, whenever you get any unidentified 2-handed weapons or salvage items, identify them. These will unlock runes and weapon upgrades for your PvP characters.