The Bay Area is the area in California, containing San Francisco, Oakland, San Jose and other urban centres. It's fairly extended, with millions of inhabitants. There's a plethora of highways, and it's probably better to see the Area as one huge city, with "The City" (San Francisco) as its cultural centre.
Going North or East[]
You can take the BART to Richmond (about 4 US$), get off, walk East, where there is the on ramp to the I-80 East. See Google maps.
To Santa Cruz[]
Go to the 19th Avenue, which is also I-1. Find a good traffic light while walking South. You can even start in the Golden Gate Park.
To Los Angeles[]
Take the I-5 if you want to get there fast. Otherwise you might better off with the I-1 and/or I-101.