Let's do this in ever-widening circles and in broader strokes to help you identify what kind of terrain you might like...
Nearest options include:
- Mount Tamalpais (north of SF, 30min away): 2600' or so peak, hiking and one or two walk-in camping sites. Gorgeous views, steep.
- Point Reyes National Seashore (north, 90 minutes). Overnight walk-in campsites, decent trail system, coastal with birds and maybe Tule Elk.
- Henry Coe State Park (south, hour or so): Often overlooked, plenty of hiking and backpacking opportunities.
These are places with backpacking possibilities. There are thousands of day-hiking trails that only a book could do justice to, from Muir Woods to forested watersheds on the peninusla south of SF to Mount Diablo in the east bay. I've been hiking in this area for a decade and have a long way to go before running out of new places to go.
Outside of this you're looking at several major areas that each also have whole books dedicated to them:
2-3 hours south you hit Big Sur and its various areas (Ventana Wilderness has already been mentioned). Coastal forests and rocky shores, tidepools galore and lots of otters!
3-4 hours east and you hit the Sierra Nevadas. Classic western mountains. Earlier poster's POV on Yosemite is spot on ... I just came down from there last weekend, nighttime temps now near freezing.
2-4 hours north and you get into Mendicino County and other areas north...Lost Coast, previously mentioned, is up that way (but farther). For reference driving to the Oregon border would easily take a full day or more. The farther north you go the more Redwoods, rhododendrons and similar vegetation you get.
Not tons more info than already given but hopefully paints the area in broader strokes and lets you determine what time commitments and what kind of terrain you're hungry for. We've got almost all of it within 2-4 hours of here... (the deserts are more like 6-10 hours away, remember SoCal's got some big wildfires down there right now)
Hope this helps,