There is no difference. There are two forms for the same verb. You can use both. ^.^
Go into means entering a room or a building, right? So I guess it's wrong because you are going to the dining room, towards it. There is no indication you're actually entering it.
Thank you. I would translate : I go to the dining room with πηγαίνω προς την τραπεζαρία but when I want to express: I go into the dining room - how can I express it in greek correctly? Πάω μέσα στην τραπεζαρία ή Πηγαίνω μέσα στην τραπεζάρια ή Μβαίνω στην τραπεζάρια
Also three that you've listed there are correct, with μπαίνω στην τραπεζαρία being the most commonly used one ^.^