Relocating to San Francisco - Need Advice on Where to Live, Doctors, Preschools

what Mamazita said. We live in the "Lamorinda" (Lafayette/Orinda/Moraga) area just east of the Oakland/Berkeley hills - great weather, excellent schools, very family-friendly, but unless you're relocating from a place with a super-high cost of living like Manhattan, you could be in for a serious case of sticker shock when you see how much an ordinary-looking 1950's-era tract house will sell for =:-O! And like Mamazita said, many of the school programs are funded by parent contributions (as well as parcel taxes and city-wide and school-wide nonprofit education funds -IIRC about 30% of our school budgets come from these sources and allow us to maintain art, computer, music, library, and classroom assistants among other things)
East Bay Pediatrics (offices in Berkeley and Orinda) is an excellent practice http://www.eastbaypediatrics.com/
Aloha Pediatric Dentistry (offices in Berkeley and Orinda) is a very kid-friendly practice - my kids loved getting checkups there but their prices are a bit on the high side if you're paying out of pocket so our kids now go to the same dentist as we do, Dr. Les Hata in the Rockridge district of Oakland (excellent dentist but FYI a very small practice - it's just him and his receptionist, no hygienists or other dentists in the practice) http://www.nvo.com/hatadds
http://www.alohakidsdds.com/

First, let me say, welcome! I live in the east bay, just a fifteen minute drive into "the city" and I love it :)
It's hard to know what area to recommend without knowing your housing budget. If you want to message me with a rough idea I could give you more specific advice (and no, I'm not a real estate agent, lol!)
The city itself as well as the area north (Marin) and south (Daly City) certainly gets more fog than the east bay (Oakland, Berkeley, Alameda.) And sadly, all California public schools are struggling with budget cuts. The best schools are generally in the most expensive areas. Parents here in Orinda raise a LOT of money to pay for things that have been cut all over the state: smaller class sizes, art, PE and music.
Like I said, PM me if you want more specific advice :)

We don't live near the Bay Bridge, but we LOVE Millbrae!! Great schools, and we love our ped, ped dentist, family doc, OB/GYN, chiropractor, and the fact that it's sunny most of the time! My husband works in SOMA, and it's a 30-min ride on the BART. We live walking distance to BART, so we only have 1 vehicle. It's worked out great, and we love that we're close enough to SF, San Jose, and Palo Alto to do A LOT of different activities :)

I'll put in a vote as a proud Oaklander - we moved here from the East Coast 13 years ago and I love it. Very diverse and hip, the benefits of city living combined with easy access to big parks (two trailheads within 10 minutes walk from my house) and a short hop across the bridge to SF.

We live in the Montclair neighborhood - if you're looking for the best Oakland public schools you'd look there or in the Hillcrest and Rockridge neighborhoods. We have loved our public elementary school, took the private school route for middle, and plan to head back to public for high school.

We definitely have our issues - crime, budgets, schools, etc. - but so far the positives have always outweighed the negatives by far. It's always 10 degrees warmer here than SF (and 10 degrees cooler than sweltering Lamorinda in the heat of summer.) And we like that our kids are growing up with an everyday awareness of both cultural and socio economic diversity. Good luck with your search!

As said in other answers, Lamorinda has great schools, weather, etc. Less fog than on the other side of the tunnel. I believe Walnut Creek also has really good schools thru HS and all of those cities are on the BART line into SF.

Piedmont, CA is a beautiful, but expensive, area that is close to SF via the Bay Bridge. Public schools great all the way through HS. Great pediatric office: http://primarypediatric.com/ They have 3 offices in the East Bay.

Menlo Park, Atherton, Woodside, Portola Valley are all on the peninsula and get great weather. The public school system is very strong there at least through 8th grade. Not all of the HS are great as I recall.

Marin County also has cities that fit this bill too. It is also tends to be expensive.

Childrens choice dentist in sf is the best! My son actually enjoys going to the dentist :)