San Francisco International Airport (SFO) is the biggest Bay Area airport. If you're staying in Berkeley or Oakland, you'll have about a 40 minute drive or an hour long trip on BART once you land.
Oakland International Airport (OAK) is a much smaller airport, but is closer to where we live: a 20 minute drive or 40 minutes on BART. The major airline at OAK is Southwest, but other airlines also operate flights to Oakland.
San Jose International Airport (SJC) will likely also be included if you search for flights to the Bay Area, but it's even farther from us than SFO (a 50 minute drive and not very accessible by mass transit: expect to take a bus or two and BART to get to Oakland). We've never flow through SJC, but if you find an amazing deal and are renting a car, it could be worthwhile.
If you're coming from Washington, it is possible to get here via train! Train tickets will require almost 24 hours of travel each way and are not significantly less expensive than flying, but the route is scenic and you'll have a smaller carbon footprint than if you take a plane!
San Francisco Bay is connected to the ocean, so there must be a way to get here by boat! But we don't know what it is. And if you're coming from the east coast, it will require either a trip through the Panama Canal or a trip around Cape Horn. Say hi to the penguins for us!