Borderlands 2 share many gameplay features from the first game, while offering plenty of new features.
You will be able to change the appearances of your characters' gear, skins and heads by getting drops from bosses and specific enemies. You will also be able to get them as rewards from quests, challenges and from leveling.
As you complete challenges and level up, you will increase your Badass rank. As this rank increases, you will earn tokens, which can be redeemed for statistics of some kind. The good thing about the whole Badass system is that it applies to all characters, even ones you haven't created yet (it's for your profile, not one specific character). Of course, the stats you can improve are randomly chosen, so you can't keep increasing the same stats.
While trading isn't technically new to Borderlands, the way in which it is done in Borderlands 2 is new. There is a new trading menu, where you can compare the item being traded, as well as the item you are trading for. Previously, you had to drop the item/weapon on the ground, and the other player had to pick it up and find it in their inventory.
Eridium is a new element that can augment other elements' powers. It is also a currency that can be used to buy the most powerful weapons. Besides those two things, it can be used to provide in-battle temporary buffs, although enemies can also use it to give them buffs (it can be shot out of their hand).
There will be plenty of quests for you to do, but some missions will provide a choice in how to complete them. For example, one mission asks you to find some "questionable" pictures of Moxxi (yes, she's returning and no, you can't see the pictures). Once you find them, you have the option to either return them to Moxxi or return them to Marcus. Some choices may even affect the storyline and the rewards you receive.