Do I have to do the skill quests to get most skills?

I much prefer just playing the missions and then buying skills from skill trainers, over doing quests from skills. I just want to unlock skills so I can go back to PVP mode.

So if I only buy skills from trainers, will I still get the majority of skills available to my class? Or are skill quests requird for the majority of skills?

Any skill you can quest for, you can buy.

but you wont get the elite skill's (you must capture them from NM's )

And finally... skills use skill points, which you may want to spend on capturing elite skills. Especially since there's such a slow down after you hit 20th.

Edit: where as questing does not use any of those.