I believe there are portions of 101 that aren't open to bicycles. But, even so, take Highway 1.
There should be no problem doing a credit card tour, though, you will need to plan where you will stay near Big Sur, which has motels but they are often booked in advance. Also, once you leave Big Sur, there are no reasonably priced hotels until San Simeon (an ugly commercial strip a few miles south of Hearst's Castle). From Big Sur to San Simeon is about 70 miles and it includes 2 good sized climbs and lots of smaller "lumps." If you are camping, consider Kirk Creek, which has no showers but a hiker/biker site and is right on the ocean bluffs. It is about 50 miles south of Carmel.
The area just south of San Francisco on Highway 1 (between Pacifica and Montera) is called Devil's Slide and is a dangerous area for bikes. I live in SF and have never ridden, though I have gone down to LA several times. I get a ride to Montera, just south of this part of the highway, and start there. As for what you can do, try posting on Craig's List for someone to give you a ride, look for a bus with bike racks, or ride to Pacifica and get a cab. You can ride it if you like but I haven't met anyone who said it was safe or a good idea.
As for going south of Santa Barbara, you will have to decide for yourself. The ride north and south of Santa Barbara is on 101, a freeway with lots of cars and trucks going 70 MPH. It has a wide shoulder but is not the kind of riding I look forward to. Once you get to Ventura, it is uninspired until you get to Pt. Mugu, about 20 miles north of Malibu and LA. The ride from Pt. Mugu down to Malibu is quite nice but it gets increasingly traffic clogged as you approach Santa Monica/LA.
Here is
my personal journal for my first SF to LA ride.
Here is a page with
40 links to bike touring in California. Not all of them deal with this route, in particular, but many of them do.
Have a great time. It is one of the world's best bike touring routes.
Ray