A Track and a Train

Public transportation in the Bay Area is a bit of a complicated and confusing process. Unlike New York or DC, which have just one primary method of transport (the Subway and Metro, respectively), there are two major options in the Bay Area, depending on the direction in which you want to travel. 

The BART is most comparable to the Metro—the seating arrangement is similar, it runs on a schedule, and the platforms have signs to inform passengers as to when the next train will be arriving. The ticket and payment operates very similarly to the DC Metro, too, in that the cost is determined by the length of your ride, rather than being a flat rate, á la the Subway. Although its name links it to the Bay Area in general, its primary use is for transportation between the East Bay and San Francisco. I've taken the BART a few times, and it's an easy process, assuming that the stations are in close proximity to where you want to go. 

From northwestern San Francisco (right near AT&T Park) down to San Jose, through what is known as the Peninsula and South Bay, the primary method of public transportation is the Caltrain. 
I took the Caltrain for the first time over the weekend, and it's a lot like the NJ Transit. There are several stations on the line, and as the train is used frequently by commuters on weekdays, there are several express trains that run from Monday to Friday. The process of buying tickets is a little confusing, though—rather than buy tickets from one station to another, one buys tickets from one zone to another. Each zone (there are six in total) comprises anywhere from two to nine stations, so you've got to look at a system map to figure out in what zone your intended destination falls before you can purchase your ticket. 

As I mentioned, these are the primary methods of transportation around the entirety of the Bay Area. But there are several more that operate on a smaller scale; the Muni tackles all public transportation in the city of San Francisco except for the BART, while most other regions in the Bay Area have access to buses. 

It's interesting, though, because while the people in this region pride themselves on conserving energy (and for good reason; there are bikers everywhere), when you really look into it, the region is not all that accessible by public transport. 

I know that probably sounds ridiculous, considering the various transportation methods I just detailed. But their reach is not as extensive as it could be, particularly considering the size of the entire Bay Area. 
The city of San Francisco, in particular, is where this void is most glaring. Once you're in the city, it's pretty difficult to get around from one region/neighborhood to the next, unless you're well acquainted with the Muni Metro or bus system—tourists, most likely, are not. Plus, even once you are familiar with these systems, they don't generally operate that efficiently, so it can be a colossal waste of time to travel this way. 
Driving is often the preferable method of getting around the city, which is problematic because it can be stressful, and because parking is next to impossible to find. 

I recognize that the tectonics of California (especially San Francisco) probably present a difficulty in building an expansive underground transportation system. And even if it were a geological possibility, that kind of system would be immensely difficult to implement at this point in time. 
But it's truly unfortunate that a city as lovely as San Francisco doesn't offer the kind of transportation access that's available in cities such as New York, London, Chicago, and DC. It'd make mobility around the city far easier.