Santa Cruz is PERFECT for mountain biking. I lived there for 13 years.
Buy this map, it is available in most of the SC bike shops.
The trails I'll mention are on this map, although there are more trails in real life than on the map. But the map will get you started.
For excellent forest singletrack, slightly-to-somewhat technical, moderately steep, green and beautful, swoopy, great traction, you will want to explore Wilder Ranch and the adjacent University of California & Graywhale areas. There are miles of singletrack in there. Many fire roads, as well.
For steeper and more tech-y terrain, you can also try the Demonstration Forest, in Aptos. (10 minutes drive south, or ride there from SC) You can drive over to the back side (highland drive) and do multiple loops, or you can ride up 14 miles of fire road, drop down the backside on singletrack, then climb out. gets you about 40 miles of riding, lots of elevation gain.
Big Basin state park is a nice fireroad loop, not technical, but quite beautiful and relaxing.
If you want a maze of tiny etched-into-the-hillside trails with a big dropoff on one side, and lots of roots, go to the peninsula (north of santa cruz, 45 minute drive, skyline boulevard) and ride El Corte de Madera and Purrissima Creek.
For big rocky technical fire roads (hot and dry) with great views, go up to Marin and ride Mt. Tamalpias, the birthplace of mountainbiking. (this is what balt_philly1 is mentioning, pine mtn loop, eldridge are good, china camp is too short to bother with, there's another one called Boy Scout Loop that requires a pass but is worth it if you love switchbacks.)
Don't bother with Mt. Diablo, too hot that season and kind of boring.
Sorry I don't know any clubs, but try the stores below.
Good bike shops:
Another Bike Shop
Family Cycling Center
Tahoe is great too, but I don't know it very well.
Make sure you get someone to show you what Poison Oak looks like, and avoid it. It grows all over there, and gives you a nasty rash.
Have a great time! I don't live there anymore, or I'd offer to show you around.