First off, I would have to say "Keep this story going." It's a great story so far and a lot of good can and will come from this. The story in Borderlands 2 is more narrative and interactive than the first one. Not to mention better laid out and executed. With a third installment I could definitely enjoy, dare I say, choices that impact the story, or a "branching" story line that changes depending on if you completed a mission or not.
Depth: While the Borderlands series has always been about shooting and looting, I think it's time for this franchise to mature a little bit. What I mean by that is: Let's have our Vault Hunters, our Classes, and our Characters feel more in depth. Instead of just having them shout "HELL YEAH" or "RAM THIS DOWN YOUR THROAT HOLE" every once in a while, let's have our characters feel real. While Borderlands 2 touched on the depth of its characters a little bit with hidden ECHO recordings in certain levels, it would be appreciated to get to know Axton, or Salvador on a more personal level. Rather than just knowing their names, classes, and that they likes guns.
More Classes: I'm not going to sugar coat this, but every class in Borderlands 2 was a "2.0" or "1.5" version of the first group of vault hunters. Salvador was Brick just with guns. Maya was Lilith 2.0. Axton was Roland 2.0, and Zero is a stealthier version of Mordecai. While there is nothing wrong with having updated versions of previous classes, but doing a "3.0" version of the same classes in Borderlands 3 would be underwhelming. Let's have some "melee" only classes or something to help spice things up a bit.
More Loot!: Getting guns, shields and grenades is fun and all, but when are we going to get some battle armor? Borderlands has RPG elements and I feel it's time for the franchise to embrace more. I don't mean just RPG elements. I mean all elements of game play within Borderlands. Let's have more cinematics. More car chases. More UNZT UNZT UNZT! Don't be afraid to embrace new things within the Borderlands franchise.
A New Experience: The first Borderlands was a nice breath of fresh air from the stagnant FPS and RPG genres of that time. Borderlands 2 felt like the same experience, which isn't a bad thing. But eventually that new and fresh experience gets old and stagnant. I guess I'm repeating myself here but 'don't be afraid to embrace new things.'
More Tiny Tina: Need I say more?
Well, I can't really think of anything else. What do you guys feel should be improved upon, changed, and/or added in the next installment to the Borderlands franchise?