Roads around SFO are a nightmare. I agree with the suggestion to hop on BART and get off at the joint BART/CalTrain station in Millbrae. Or ride either train service all the way to San Francisco if your aim is to ride from San Francisco to L.A. or something like that.

San Jose isn't nearly as bad as SFO. When flights arrive traffic is bad, but dies down pretty quickly. Rather than attempting to use the car-clogged main exits from the airport (Airport Pkwy, Skyport Dr), I would stay right on Airport Blvd and loop around "behind" the airport toward Coleman Ave and De La Cruz Blvd. Heading north, Central Expressway is heavily traveled but has a very wide shoulder once you're away from the airport. There's generally much less traffic on weekends, especially on De La Cruz Blvd.

If you don't want to deal with Airport Blvd, and I wouldn't want to, you could probably catch a shuttle bus to the short-term parking lot which is located behind the airport along Martin Ave (just off De La Cruz Blvd). You can also use the public transit busses to transfer between the airport and CalTrain or VTA Light Rail. I can't remember if the VTA Airport Flyer busses have bike racks, but the busses don't seem to be used much so I would think you could probably bring a bike inside if they don't have racks.

In terms of shipping, the only service I've used for my bike is Amtrak. It's extremely cheap ($15 for LA->SF) if you're traveling with your bike. Sending just the bike is more expensive but still reasonable as I recall. I've had good luck shipping larger, though never a bicycle, items via Greyhound Package Express. Items need to be well-packed in order to survive the experience; bus drivers aren't any more gentle than airport baggage handlers. They don't do door to door delivery. You drop your bike off at a Greyhoud depot and it must be retrieved from the destination depot. The receiving depot will call you when your item arrives, so you can ship your bike out a before you leave and be sure that it's arrived before you board a plane.