I do them all on my first class. You're going to end up FATE grinding anyway, no matter what you do. Plus you get an exp bonus from leveling any class that's under your highest level class.

And honestly, while clearing an area of quests does get you a nice exp boost, you can also sometimes get just as much of a boost from doing FATEs in the area. For instance, I was in Coerthas last night and three times did Svara, which is a three stage FATE. The first stage I got over 7K, the second stage over 10K, the third stage over 20K. That's 37K exp and it happens faster than doing the same number of quests I would need to get that total.