We have been to Sydney before. We would like to stay in a neighborhood this time. A neighborhood where we can feel less like a tourist would be great. I really would like this neighborhood be close to transportation to downtown and other sites, have a good bakery and coffee place and maybe some good casual restaurants. Would this neighborhood be Potts Point, Surrey Hills or any other areas?

It could be, and it might also be Kirribilli, Cremorne, MaMahons Point, Balmain, Glebe, Erskineville. Get a good map of Sydney & look at a radius of 5-6 kms from the CBD.

Assuming you're looking at Airbnb / an apartment - the more residential, the fewer hotels, as you would no doubt expect.