Here's where to get started with CFS; California Floaters Society (CFS) - a community of California river rafting, kayaking, canoeing and river boarding enthusiasts
Once you're on the Google group you can post to the listserve, and respond to other's posts. There are several members with kids and occasional (annual?) camping trip on the South Fork of the American (SFA) There are more CFS trips on a fork of the American, east of Sacramento, than all other rivers combined.
Gold Country Paddlers (GCP) is pretty much 100% kayaking. They have an annual camping trip to the Trinity, on the 4th of July weekend, IIRC
For day trips with your little ones nothing beats the Kings, and it's only an hour from Fresno. But there aren't a lot of play spots for the kayak, and only about 6 class III rapids in the 10 mile run. It's boatable right now @ 1000 cfs, which is minimal for a raft.
The Merced is 2 hours away, and gives you the oppertunity to up your game with the kayak. It's class IV.
As for overnighters with the kids; there's primitive camping on the Kings, but if you want to load the camping gear in the raft and stop somewhere on the river you almost have to leave the state...unless you want to break up a 10 mile run into 2 days. The most popular 3 day wilderness run for californians is the Rogue in OR.