I grew up on the rivers down there. There are tons of great runs and the weather rocks! The closest class III to San Jose would be in the foothills of the Sierras, directly east, from 2 to 4 hours away. The American has some very fun and popular (read: crowded!) runs. When New Melones reservoir is low enough the old Camp 9 run on the Stanislaus is great, except for the bathtub ring left by the reservoir. The Toulomne has an awesome class IV run that is one loooonnngggg day or a nice overnight although you may need permits for that one.
There are some rain fed runs a few hours north of the Bay Area as well.
The cacreekin.com website in an ealier post has tons of great runs to choose from and good beta, even though it was developed for IK'ers. Boof.com is good too but seems to be more for hard core boaters.
I would suggest picking up a copy of "The Best Whitewater in California" by Chuck Stanley and Lars Holbeck. This book is the CA bible. It appears you can get a copy pretty cheap on amazon...
Just over the hill from San Jose in the Santa Cruz mountains, the San Lorenzo runs on big rains but beware, it tends to pick up, um, "floaters" from overworked septic tanks. It can also have some serious redwood strainers so be careful.
There is some great surfing along the coast near Santa Cruz and down into the Monterey Bay but beware of two very scary creatures found at many of the breaks:
1. Santa Cruz surfers (they hate kayakers and will hassle you big time).
2. "The man in the gray suit" (largest great white sharks in the world, and lots of them).
Have fun!