SFO has quite a few European carriers with nonstop flights to their home countries.
Off the top of my head, I've seen British Airways, KLM, Lufthansa, Virgin, Air France, and I think (don't quote me, please) but maybe Alitalia?
American also has a very large presence in SFO following their acquisition of Reno Air a few years ago. They have flights to dfw, ORD, mia (I think), all the major west coast destinations (sans Seattle and Portland, maybe.) And there are flights to New York and Boston I do believe.
Besides the largest presence of United, and a sizable presence of American, all the US majors have flights to their main hubs from SFO. But, depending on where you live in the Bay Area, SFO may not be your closest airport. SJC is a very nice facility (well, if you're on American or Southwest) with a good array of flights.
San Francisco is a great city! Hope you enjoy living there.
live forever and stay beautiful,
"[He] knew everything about literature, except how to enjoy it." - Yossarian, Catch 22