Oh, man there are so many places you can go. First, it depends on which mountains you’re talking about. There are mountains north, south, and east of the bay.

The coast ranges provide some of the most stunning views and driving along Hwy 1. Here is a gorgeous ocean-view place to stay 5 hrs south of SF. You can drive along the ocean-front highway most of the way:
http://www.treebonesresort.com/

North of SF, Mendocino has tons of B&Bs. Just Google it and you’ll find tons. Before Mendocino is Stinson Beach, another great sleepy town. A drive around Tamales Bay and Point Reyes is one of my favorite things to do.

Napa county is pretty mountainous compared to the midwest, so do consider it. Calistoga and Sonoma are nice, more secluded towns near Napa. These are probably the best mix of close and awesome. (1 hr from SF)

In the east the Sierras offer tons of places. Grass Valley is a cute gold rush town with B&Bs. If you go far into the Sierras, though, you might hit snow this time of year.

You can just google all the places I’ve mentioned or look for B&Bs at their city sites. Have fun!

EDIT: I forgot to tell you about one of my favorite places in the world! A thatch-roof inn with a cozy restaurant, pub, and game room for guests downstairs, with elizabethan-style rooms, four-poster beds with tapestry curtains, balconies and full english breakfast delivered to your room. the beach is at the end of the lane and it’s all surrounded by muir woods, the famous redwoods of CA. stay there!!! (1 hr from SF) http://www.pelicaninn.com/

EDIT: Oh yeah, here’s another option for staying on the coast. You can get private rooms at this lighthouse hostel. I stayed there once in the communal part. It was nice. The private rooms might actually be in the lighthouse (?) you can call and ask. They have a little campfire by the lighthouse on the cliff and it’s so amazing to sit at night getting cozy with the roar of the waves below. Check out the photo gallery. It’s gorgeous (~25 mi south of sf) http://www.norcalhostels.org/montara/

Here is a resource for more lighthouses in case that one’s full or anything: http://www.rudyalicelighthouse.net/CalLts/CalLts.htm

You might want to take a look at the Evergreen Bed and Breakfast, Pleasanton, California, 45 minutes from San Francisco. Bills itself as an ‘intimate hiding place’, a relaxing weekend retreat, an unforgettable honeymoon…next to the Augustin Bernal Park and Pleasanton Ridgeland, with hiking and biking trails criss-crossing the area, plenty of greenery and a natural landscape.

9104 Longview Drive, Pleasanton, CA 94588; (925) 426-0901.