When you kill and enemy you need to have skills on your skill bar, this dictates how the exp is issued. So make sure you have skills for each skill tree you want to level.
Ok, so if I want to level Resto Staff, while using a Destro Staff I would just need to have some Resto abilities on the bar? Does your level affect the amount of the xp, or is the skill level of the weapon? I was doing Fungral Grotto last night and healing, and my resto staff barely moved. I'm only level 15 though so I feel I should have gotten a lot more experience for this.
Another way is, if you can remember to, keep resto on weapon swap, and swap to it and a full bar of resto skills before turning quests in and the xp from the quest will go straight to your staff.
I also noticed that the lower level skills don't take as much xp to rank up so if you are really high level and do anything with a new weapon those level in a couple kills.