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.
Note that because the Bay Area is so spread out, just getting around within it can cost significant cash (exacerbated by the distance-based fares charged by the BART and Caltrain rail systems, and the fragmentation of the bus system between different agencies in different parts of the area). Getting from one end of the area to the other can cost $12 to $15. But at least within SF itself it's not so bad, walking between most points in the city is pretty feasible and the $2 Muni bus fare covers unlimited rides for 2-3 hours.
Going North (US-101)
From San Francisco, the best place to hitch in this direction 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 couple blocks down that angular street (at Richardson and Lyon) is a large rarely-used bus stop area which makes a perfect hitching spot.
Immediately across the bridge is Marin County, which is mostly a rich suburban area where getting a ride is difficult, so if you're going up 101 it's best if you can get a ride from SF at least as far as Novato, and preferably beyond Santa Rosa. Between Novato and Petaluma is a stretch of non-freeway where you could hitch on the road, which is probably better than hitching on a ramp if you want a long ride north towards Humboldt/Oregon. (check the map for this spot) Alternately you can take Highway 1 from Marin County up the coast, but while very scenic this road has little traffic so be prepared for long waits.
Going Northeast (I-80)
You can take the BART to Richmond (about $4.50), get off, walk East, where there is the on ramp to the I-80 East. See Google maps. Waiting time at this spot was half an hour for a straight ride to Sacramento, April 2016.
You can also take BART to Pleasant Hill (about $5.25), 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.
There is a tradition of "casual carpools" across the Bay Bridge (check that website for further info). This is a good way to get out of the city along I-80 if you're leaving in the afternoon, but less useful in the morning as it's mainly for commuters. Some but not all drivers ask for a $1 contribution (still cheaper than BART).
You can try to follow I-80 to get to I-5 north, but if you're trying to go north you'll probably have an easier time on US-101, see above.
Going South (Highway 1, towards Santa Cruz and scenic route towards Los Angeles)
In San Francisco, go to 19th Avenue, which is also Highway 1. Find a good traffic light while walking South. You can even start in the Golden Gate Park. Along Great Highway (e.g. at the end of the L metro line) also works extremely well and may have less local traffic. These are both very good options if you don't want to pay for a bus/BART.
You can also take SamTrans bus 110 for $2 from Daly City station (just south of the SF city line, reachable by BART or Muni bus) 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.
Continuing beyond Santa Cruz towards Los Angeles along Highway 1 will probably be slower than taking I-5 but this is considered one of the most scenic roads in California.
If you're starting from closer to San Jose, you can also reach Santa Cruz by taking the VTA 68 bus from there south to Gilroy for $2 and hitchhiking out on 1st street, which is also highway 152 (west); this is a good hitching spot in the morning when people are headed from Gilroy to Santa Cruz to work. Though at that rate the 17 bus from San Jose all the way to Santa Cruz is $5 and takes a much more direct route.
Going South (I-5, fast route towards Los Angeles or Las Vegas)
Take the I-5 if you want to get there fast. You might want to take the BART train to Dublin/Pleasanton (about $6). The BART station is located next to a highway onramp, but you might have better luck if you walk further East to the following onramp. (You're actually hitching onto I-580 here but it merges into I-5 further out.)
This route worked really well for an anonymous user in 2009: Take Caltrain south to San Jose (about $9 from downtown SF, $7 from the BART transfer at Millbrae). Take VTA bus 68 to Gilroy from there, 1.5 hours, $2. Hitchhike east along the Pacheco Pass highway (152 east), which leads towards I-5; many cars here will be continuing south to Los Angeles on I-5. If heading towards Las Vegas, get off I-5 at exit 257, to get onto highway 58. Take that East, avoid getting stuck in Bakersfield. 58 goes all the way through, past Mojave and Barstow, towards Vegas.
A note regarding Caltrain: tickets often aren't checked, but fare inspectors do come through sometimes (if you ride end-to-end you have maybe a 1/3 chance of seeing one) and if caught without a ticket the fine is nearly $300, so think twice before risking it.
If you're considering taking a bus: booking months in advance you can sometimes find Megabus tickets from San Francisco to Los Angeles for as little as $1.50, though it's usually at least $10. With less advance purchase, Megabus and Greyhound are typically in the $20-$40 range.
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.
San fran has a lot of great services. The Larkin Street drop in center is open monday through friday from 11-2. You need to be 24 or younger but people lie about their age all the time and they seem to not care if they find out. They have a ton of food, and numerous services, and can help you get bus tokens, P38s (can openers) socks, hygiene items, etc.
The HYA (Homeless Youth Alliance) , who walk around Haight every day, are great for grabbing some snacks, condoms, or hygiene items. They run the needle exchange (monday, wednesday, friday 5-7 across the street from the good will on Haight and Cole st) and will hand out snacks etc there as well. If you have a habit, they give out free crack pipes, and can help you with draining an abcess.
Homeless people in San Francisco can get pretty much any Quality of life ticket thrown out by the DA. ask the Larkin Street Haight drop in or the HYA (Homeless Youth Alliance) for how to exactly do that. Other Free places, see : Chowhound or Free Charts