Hola omgitzme,
Your planning is perfect (seriously!), except there isn't a train to Santa Cruz. If you're seeing a way to make a reservation, you're seeing a train-bus connection.
I would recommend you take BART (Bay Area Rapid Transit) from San Francisco to Daly City, and ride from there, or just ride from the city. You can take your fully loaded tour bike on BART, just roll it on with you, no boxing or other BS. Costs about $5-10. From Daly City, you can ride to the coast in an hour or less and be on the Pacific Coast route at Pacifica. There's a hostel in Pigeon Point, which is a lovely location, and I hear it's nice (haven't stayed there) - if you want to break up your trip to Santa Cruz.
If that doesn't sound good, have your bike shipped to a shop in Santa Cruz and train/bus to SC from SF without your baggage, because otherwise it's going to be an insane hassle. You could easily site-see in SF without a bike, as it's got the best public transportation on the west coast.
I did Daly City BART to Santa Barbara in 7 days, without site seeing, and I'm an experience tourist and avid cyclist - so your timing sounds good.
I recommend you pack light, as there are plenty of services.
Have fun!!!!
Edit - try to camp at Kirk Creek campground in Big Sur - few services, but great view.