For info on the neighbourhoods, check out this link on SFGate:
http://www.sfgate.com/traveler/guide/sf/neighborhoods/

I think some of these 'names' are only used by the site, for ex., I've never heard anyone actually use the term 'deco ghetto'.

For apts, check out www.craiglist.org

I just did a search for 3 bed apts for $3250 in SF...and I found several options, mainly in more outer neighbourhoods, but onese that probably have more local flavour (as you have expressed a preference for).

Be aware that you won't have parking for 3 included at that price and you may end up having to pay for parking...which could add several hundreds dollars to your cost. Or you may have to search for parking...