Exploring Richmond, looking for neighborhoods to live, amazing restaurants, must do/see things before deciding where to live. Please help!

My SO and I are here for the weekend from Raleigh. We're 90% sure we're moving here in a month. Please help us discover Richmond so we can figure out where to live. We are mid thirties, no kids, dog owners and just drove around through Cary Town, The Museum District, and Fan (hope I got those names right), and immediately fell in love. Any suggestions would be GREATLY appreciated.

EDIT: Realtor suggestions would help a ton, too! And spelling