Climbing is a very popular sport in the Bay Area, with (at least) 4 major indoor gyms and considerable choices outdoors. I'm of a large group of people who do it entirely on the indoor walls - it's easy, it's cheap, and it's still a blast.
Required gear: climbing harness, belay device (ATC, though Planet Granite uses GRGRs on all their ropes so you don't need your own) climbing, shoes, chalk bag. All can be rented at the shop. Clothing - long sleeves and pants protect your skin from scrapes, but may be too confining for some. Gyms can be a tad chilly though. I like to go in a tank top and running tights. I tend to sweat a bit.
The gear can be purchased from the gym, or from places like REI relatively cheaply - 175-250 would be typical range. Everything but the shoes are sometimes bundled for 50-90$ in starter packs. For indoor climbing, nearly any comfortable harness will do the job. Padded leg straps are good. If you plan to do more serious climbing, more thinking should be done. I can only suggest talking more with your instructor/friends who do that.
Requires time commitment - half day typically.
Costs - gyms charge 8-20 dollars for access, and rental of all equipment
is on order of $6. An intro with the necessary safety training for
belaying is bundled with admission cheaply - $25-30 for everything you
need for your first day of climbing. If you're not with someone
experienced, you'll want this, or the more formal training of a rock
Inside the gym are walls with many ropes dangling down. You pick a route you want to do and tie into that rope. Your partner attaches the belay device at the other end. Routes have a rating system that ranges from as low as 5.3 up to 5.13 (or up?). 5.3-5.5 resemble climbing a ladder - you go straight up and there are lots of good holds. 5.6-5.8 is the beginner zone where you stay until you get stronger and better skilled. If you go irregularly, it's hard to get past this level. Some holds are hard to grip, and you must leverage your force in other directions. Above that, you have to jump and other fun stunts. Mission Cliffs and Ironworks have a lot of routes that slope backwards. Getting over these elements can be very difficult and make those gyms a bit harder.
If you want to do real climbing, there should be many options for you. The only one I know is with Cal Adventures who does a few levels of training and then trips. The Berkeley Hills and the Pinnacles of Hollister are among the local spots, with of course all sorts of great climing in the Sierra.
Planet Granite -
located at Santa Clara, and a new one recently opened in Belmont.
Touchstone Climbing Centers - located at Berkeley (Ironworks), SF (Mission Cliffs), San Rafael (Class 5).
At each of these, membership can allow use of the other gyms in the system. Planet Granite offers discounted rates for members of Touchstone. I've been to Ironworks and Mission Cliffs and if you have a choice, go to Berkeley. A bit bigger, and less crowded.