Family rafting rocks!! I heartily agree with much that's been said, and I have a few additions:
1) The Trinity has enough water to raft all summer (due to an old lawsuit). Anything above 600 cfs, and you can still get splashed, plus there is enough water to miss the more technical challenges. 2,500 and 4,000 cfs are my favorite levels for bouncy, splashy, Class III fun. See the release schedule at: http://www.trrp.net/restore/flows/current/
2) A way to have your kids fall in love with Class III: Find long wave-trains of Class II (deeper, no rocks) and swim them on purpose with your kids on hot days. Make it crazy fun, and if your kids like competing, set prizes for catching eddies, etc. This has two long-term benefits. You will learn A LOT about maneuvering your boat from swimming rapids. Your kids will learn that is is normal and can be totally fun to swim in whitewater. That way, when they do fall out (or you do again), nobody feels freaked out. Said wave trains can be found this year on the SF American, Trinity, Rogue, and likely the Kern (all within 7 hours of you).
At the risk of reiterating, kids love rowing, not paddling. If you like camping, then I highly recommend building/obtaining a frame and some Carlisle oars.
3) A great first multi day trip that's really close to you is the East Fork of the Carson. It likely won't have water this year, but its totally worth doing above 1,000 cfs. Its 10 miles of fast water with some Class II to a GREAT hot spring (camp two nights there), then another 10 miles to the take out. We do it on long weekends from Redding.
How we spend our days is, of course, how we spend our lives.