Before BD I actually thought Bella's dreams would somehow be related to her power when she became a vampire, because they were so vivid and relevant to what was going on in her life. I thought they would translate to some kind of visions, but of course be different from Alice's.
I think Bella must have been a vivid dreamer before moving to Forks because she mentions that she's always talked in her sleep. I associate that with deep dreams but I could be wrong. I do think the tone of her dreams definitely changed when she got to Forks though, with all the supernatural activity fueling them.