Well I'd recommend getting one of the characters and Tiny Tina. Scarlet DLC is nice for the sandhawk. But i'd go with whatever is on sale first. Idk which DLC are on sale right now. I know the psycho is though. And Tiny Tina might be.
Best thing to do there is imo to actually watch build videos on youtube. To see different builds for characters it'll really display how they work and from there you should decide based on what you like.
I like this guys builds he does a good job at explaining everything. Idk a build for Gaige, I never looked into anything for her though. But I'm sure you can find good ones by searching.
Krieg is typically Elemental damage or melee damage based and should be up close and personal a lot. Gaige uses a lot of shock damage and has death trap which can buff your shields and other things. But you don't have to rely on her action skill that much though with the right setups. So just look into them.