To add to this, quality has to be quantified, and that's tougher to do. I've never been to New York, but I'm sure that's still the best.
However, I do live in DC and was raised in San Francisco, and I can say without a doubt that DC is light years better.
Here is DC, we have the Metro, an expansive bus system throughout the DMV area, a light rail system between DC and Baltimore (MARC), our streetcars are due to open this year (thank GOD finally on H St.), and we have Amtrak. NOVA has started the process of a new kind of street car just this month as well that would essentially be a bus with dedicated lanes and control over the traffic lights system for uninterrupted travel. All of this flows quickly and smoothly, so much so that people are actively involved in making it better regularly, and minor issues are a big deal that people address. That's nice.
There are of course issues, but compare to SF. In San Francisco, we have BART. BART (Bay Area Rapid Transit) runs through the Bay Area in a smattering of the suburbs and throughout the city. It's terrible. Trains are, firstly, extremely old, old enough to regularly cause problems. They're slow as well, physically and in their ability to reach stops. Half hour waits between trains aren't uncommon. On top of this, they treat their employees so poorly they go on strike on a regular basis. It's also massively overcrowded. The bus system is an absolute worthless nightmare. Nearly impossible to predict, overcrowded, and slow, with terrible routes that take far longer than they should. Their one good system is the MUNI, their light rail, and while I didn't use it much, it was smooth and quick, it was always massively overcrowded. Many times I used it, I had to wait three cars before being able to get on. And that's only in the city proper. If you're out of city limits, good luck doing anything without a car.
Of course the west coast is more spread out, especially the bay area, making it especially difficult to coordinate these things, but it doesn't change the fact that outside commuting on BART to the city, it's effectively useless.