Further to what Table 5 said, yes, you will simply not be able to afford to live in Palo Alto and Menlo Park. Not saying that there aren't perfectly nice spots nearby, but those are for the elite of an already wealthy area. San Rafael/Marin County is also likely out of the question.
I would suggest you stay away from anything in the East Bay that isn't Berkeley, and only live there if you want to be a hipster or flat out hippie.
Just about everything on the Peninsula is nice, but the commute will likely involve the 101 which can be problematic at the best of times. We live in Foster City and very much enjoy it. It's quiet, safe, and has nice places to live, but it is still kinda expensive.
San Mateo is ok, but all of Redwood City, Belmont, and Burlingame are wonderful little communities.
In the city itself; there are plenty of characters and dirtier areas, but most of the city is really nice. Noe Valley and Sunset are nice, Richmond is nice but bordering on pricey, and The Mission is cool if you want a livelier scene. Russian Hill is another good choice. Sea Cliff, Pacific Heights, and Marina are gorgeous but similar to Palo Alto in terms of price. The Castro is the gay area if you are of that persuasion. Stay away from the Tenderloin at all costs. SOMA (South of Market) is good if you are single and are ok with a smaller apartment. Really you can't go wrong but pricing is high all over the city and the culture is pretty good all over. Really fun city if you can afford it, but parking and traffic are almost a constant headache.
Make sure your place has a parking spot or a garage. You'll thank me later.
San Jose has some nice spots, but there is also some violence problems there too, so make sure you're in a good community if you decide on that as an option. I'm not familiar enough to say what is good or bad there.
Best of luck to you, and welcome to one of the best places to live in the world!*
*If you can afford it.
