The Bay Area is an 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.
You can also take BART to Pleasant Hill, there is an onramp next to the BART station, but the best onramp to hitchhike at in this area is about a mile north at Monument Blvd. It has a nice stop light on the on ramp itself. A sign saying "Sacramento" or wherever you are going helps.
For going north from San Francisco, best place to hitch is on approach to Golden Gate Bridge from Lombard Street. Just before it reaches the Presidio, it makes an angular turn toward the bridge, and a block or so down that angular street is a perfect hitching spot.
To Santa Cruz
From San Francisco, Go to the 19th Avenue, which is also Highway 1. Find a good traffic light while walking South. You can even start in the Golden Gate Park.
You can also take SamTrans (it's $1.50 from Daly City, $4.00 from downtown SF) to Pacifica, and get off at Reina Del Mar and Hwy 1. Much more low key than dealing with the city, if you have the spare cash.
Once you're out of the city and dealing with smaller town traffic (Pacifica and South), getting picked up is a piece of cake.
Take the CalTrain (it is easy to get on w/o paying though inspectors do come through) all the way south to Gilroy. Hitch hike out on 1st street, which is also highway 152. Good hitching spot in the morning when people are headed from Gilroy to Santa Cruz to work.
To Los Angeles
Take the I-5 if you want to get there fast. You might want to take the BART train to Dublin/Pleasanton. The BART station is located next to a highway on ramp, but you might have better luck if you walk further East to the following onramp.
Otherwise you can go through Santa Cruz and take one of the most scenic roads in california and take Highway 1 and/or US-101. (see above)
I5 South, and on to Las Vegas
Just done this route, worked really well. Take BART south to Milbrae. Change onto Caltrain system, take that to San Hose for maybe 7 dollars. Take bus 68 to Gilroy from there, 1 hour 20 minutes, 1.75 dollars. Find Pacheco Pass highway, head east with that onto I-5. South on I5, come off at junction 257, to get onto highway 58. Take that East, avoid getting stuck in Bakersfield. 58 goes all the way through, past Mohave and Barstow, towards Vegas.
Things To Do
The Bay Area is a diverse place, with a long and proud history of Left-leaning politics and Counter Culture. It is also, as a whole, one of the wealthiest and most expensive places in the United States. The area should have something to offer for everyone.
The Long Haul Infoshop in Berkeley, is an anarchist library and community space which often has neat events.
In North Berkeley is one of the worlds best known punk rock clubs 924 Gilman Street with shows every Friday and Saturday, and sometimes Sunday.
The Bay Area is home to a number of great parks for relaxing and hiking in. Golden Gate Park in San Francisco is a lovely urban park. For hiking it is best to go further out to Mount Diablo State Park in the East Bay or south to Big Basin Redwoods state park (or walk across the Golden Gate Bridge turn right walk down the steps under the bridge to the other side walking north a few more yards to Conzelman Road, walk up and West through the Marin Headlands, ).
The Bay Area is also home to a number of sports teams, including two american football teams, the 49ers and The Raiders.
Food Not Bombs serves free meals both in San Francisco and Monday-Friday in Berkeley (at People's Park between Dwight & Haste, above Telegraph Ave.) and is a good place to meet people.