Not sure if this was already spelled out or not... but here it goes anyways..
You can get flight following without filing a flight plan... and vice versa.. those two are not tied at all
As for SoCal and Bay App..
SoCal = Southern California Approach. They handle most of southern california, hence the name.
Similarly, Bay App handles the San Francisco bay area.
Both of these are really consolidations, so you don't have LAX approach, Orange county approach, and so forth.. there are so many (big) airports in a relatively small (air)space that they simply consolidated together. Same thing for SFO, OAK and other airports in that area.
Don't be surprised if you start wondering "why the f*** can't we have the same GA system in Europe" after your trip
As mentioned, the GA here is light years ahead of anywhere else.