You are in for a real treat, from Monterey on down to San Luis Obispo. ACA has maps which lay out a route for cyclists to follow - Map 4. It starts at the Golden Gate Bridge and ends in Santa Barbara.
San Jose is a large town (9th or 10th largest in the country) and spread out. If you are planning on rolling out of your driveway, where you live might influence your choice. As you well know, the problem is the lack of easy roads through the Santa Cruz Mountains to the coast. I was hoping someone would chime in and tell you the easy way to get there. And, maybe, someone will.
In the mean time, I will share with you what we have found as far as possible routes, moving from south to north:
Gilroy - Hollister - Watsonville - The idea would be to ride down to Gilroy on surface streets, use US-101 to get to Highway 129, and ride to Watsonville. The problem here is that, it seems, US-101 does not allow bikes on it. We have driven US-101 south from Gilroy, specifically trying to imagine bikes, and have seen "No Bikes" signs. We have never seen a bike on this part of US-101. There are several narrow bridges, with no shoulders, poor sight lines, and one is quite long.
An alternative would be to take a long detour by going to Hollister, using Bolsa Road - Highway 25 - go nearly to Hollister - Highway 156 - back to US-101 - Highway 129 to Watsonville. I am not sure about Highway 129. Parts seem narrow, poor sight lines, difficult, and might be startling for drivers to find a bike on it. But I am sure a rider would be fine on it.
Hecker Pass - The idea is to ride down past Morgan Hill, and take Highway 152 over to Watsonville. We have never ridden this, either. It is quite steep, very busy, has no shoulders, and poor sight lines. If you consider this, go drive in your car. We decided it is too dangerous on a loaded Surly.
Old Santa Cruz Highway - The idea is to get to Lexington Dam, ride up Old Santa Cruz Highway, ride Summit Road over and take Soquel San Jose Road down to Soquel. Once at Soquel Avenue, a cyclist is on the ACA Route.
The issue is getting to Lexington Dam. The only way I know to ride a bike to it is up the dirt section of Los Gatos Trail, complete with an impossibly steep little hill for a loaded Surly, not far from the face of the dam, as well as the dam face itself.
A cyclist could start at one of the pull-outs on Old Santa Cruz Highway, just south of Bear Creek Road, and avoid the dirt section. In fact, this is the very route my wife and I take.
Highway 9 - The idea is to ride to Los Gatos, climb to Skyline, and then either go on to Santa Cruz on Highway 9, or take Skyline to one of the roads down, like Bear Creek Road, or Mountain Charley Road. We have never ridden loaded Surly's up Highway 9, but imagine it to be a long, steep slog. Once at the top, the rider still has a long way to go to get to anywhere.
Other Northern Options - I am sure you have looked at the map, and wondered about Page Mill Road, which would involve a lot of climbing, and still be no where when at the top. Highway 92 to Half Moon Bay seems dangerous, due to the very high volume of traffic, lack of shoulders, and sight line issues.
Starting in San Francisco - A cyclist could take surface streets up to San Francisco with little risk. We have been dropped off in San Francisco before, and followed the ACA Route. On the ACA Route, Devil's Slide has been slightly eased by the opening of the tunnel. I'm not sure if bikes are allowed in the tunnels, or if a cyclist still goes over the top like we did last year, sans the cars. Non-car ways of getting to San Francisco could involve riding up on CalTrain or BART, but you would need to confirm the logistics, timetables, and costs before you attempted this.
I am hopeful I am over thinking this, and a simple, clear, easy, safe choice exists. I am also hopeful that, over time, ACA will provide recommended routes from air ports / train terminals in large US cities (such as San Jose - which is surprisingly much larger than San Francisco) to the routes. Until they do, we will continue to take Old Santa Cruz - Soquel San Jose Highway to get to the coast.