Center City, Old City, Northern Liberties, Pennsport, Art Museum, Graduate Hospital, Italian Market, Fishtown, Rittenhouse Sq, Fitler Sq, Society Hill, Avenue of the Arts, Passyunk Sq...
We really love Philadelphia.
Good luck and welcome!
Philly is so big that it really depends what your looking for in a neighborhood: proximity to downtown, established v. up and coming, amenities in the neighborhood, etc. A good starting point is to tell us what your price range is and what's important to you in a neighborhood.
Philadelphia offers all things to all people and is a wonderful place to live in my opinion. You have a variety of areas to choose from depending of budget, lifestyle, proximity to your job, etc. Please refer to my website http://www.liveinphilly.net to see descriptions of Philadelphia neighborhoods