Had a fun game this evening, got from A Dam Fine Rescue to completing Bright Lights, Flying City with Maya, level 15 now which is one level below where I should be for the quest I was on, so I'm probably three below the next quest. If I remember correctly that's the Hyperion wildlife preserve so I should probably level up before starting that quest.
I've only used four of the classes enough to have an opinion of them so far. From best to worst I'd say the order goes: Zer0, Gaige, Salvador and Maya. I imagine Axton will slot in between Salvador and Maya. I am probably biased towards Zer0 because he's the only one I've gotten to level 50 and so has a lot of skill points invested, I imagine when I've done the same and worked out decent skill sets with the rest I might change my mind. As it is, though, Zer0 with the Law and Order even with only some of the skills I've given him is incredibly good (I know that because Salvador and Gaige put the Law and Order to good use, so Zer0 would be even better with just Backstab and Be Like Water, for instance).
Seriously, though, having a powerful melee attack basically means you'll always have an awesome (but limited) weapon since it scales with you, unlike any other weapon in the game. I could even be using a Law from normal mode, but since it's a quest reward from a seriously easy quest which takes all of about 10 seconds there's not much point in that.