It's not compulsory to do quests, but I think it's a good idea to do them.
1. You get sexy exp on later levels.
2. You get awesome $$ from them as well.
3. You get fame, which you MIGHT want if you need alchemy stones for material composition/decomposition.
There is a list
here but it is not complete.
But yeah.. at about.. I dunno, maybe level 25+, the quest rewards get better. Plus, there are some quests that are in a series (i.e. Quest name 1, Quest name 2 etc) and they usually give awesome exp and money rewards. Besides, you're bound to grind some time during your levels, so might as well do some quests while you're at it. xD