I live in the East Bay and I'm a Berkeley student. I was looking for some fun rides and ways to get across the bay. Unfortunately, there's no way across the Bay Bridge. Here are a couple I've thought of:
BART - easy and runs all the time
Ferry - Never tried. Looks pricy, but scenic and fun.
Transbay Bus - Free! (because of Berkeley class pass)
Carpool - Never tried. Hard to work out times on weekends and random times.
Take the transbay bus. It's free cuz you're a cal student (GO BEARS!!). If the transbay terminal isn't where you'd like to end up, you'd be better off taking bart across and popping out closer (hopefully) to your destination.
But you're on a bike. Take the bus. You can bike a few more blocks.