Recommendations for good areas in Oakland and also East Bay in general, with nice access to the city through BART?
Alameda or Jack London Square, and you can hop on a ferry.
BART gives you 5-15 minute headways from 5am until midnight/1 am, and is very, very reliable except when it isn't. Ferry service is less frequent but generally a nicer ride (more room, less riff-raff, much less scary crap in the upholstery, literally).
If you're ever near the Oakland Tribune Tower, ping me and I'll buy coffee/beer.