The MOST important quests to not miss out on are your Black Spirit quest line and the ones with Storage Expansion rewards!

Typically, you don't get any leveling XP from quests though - just killing mobs/grinding is the fastest for that. You will get some leveling XP for crafting/fishing apparently as well, but not as much as can be earned in grinding masses of mobs.

If you do want to trade/craft/etc you will NEED to do the quests to unlock nodes with contribution points. That is the only place you get CP (Contribution Points) from.

Taxman

"Any group is weaker than a man alone unless they are perfectly trained to work together."
- Robert A. Heinlein, Starship Troopers