I spent the first five years of my kayaking in Reno, NV (right about the time the WW park went in). The park itself is a great place to hone your skills.
There's some info on runs out there at California Creeks - Whitewater Rafting and Kayaking
. Class III that's not too far from Carson...
1: South Fork American (Chili Bar and the Gorge) - Chili Bar gives you more Class III bang for the buck with one class V-. The Gorge section has more flatwater and can be especially painful if the lake level is up, but is definitely worth doing for more Class III. Driving south side of Lake Tahoe, you're only looking at 1.5-1.75 hrs drive over, too.
2: Truckee River - the section from Truckee to Floriston is mellow Class II-III, then picks up from the Floriston section starting with Jaws/Bronco (IV ish depending on flows) into Verdi. There are three dams en route to NV-all are runnable but it's best to go with someone familiar with them the first time to get the lines.
3: Cal Salmon (Otter Bar Lodge) - it's about 6.5 hrs from Carson, but well worth the trip for some instruction. Absolutely awesome place to learn/improve and well worth the $$. They have everything from ponds at the lodge to Class V with everything in between. Learn To Kayak | Call Today! 1.530.462.4772 | Otter Bar Lodge Kayak School
After you've graduated to Class IV, you'll love...
4: NF Feather River (Cresta and Rock Creek) - monthly releases in the summer and a river fest at the year end release. One of our all time fave Class IV rivers (about a 2 1/2 hr drive from Carson). I still trek back from CO for the releases on occassion.
5: NF Yuba - Goodyear's Bar
6: Chamberlain (American)
Some helpful sites include:
Dreamflows Home Page
boof.com, the california whitewater community - Powered by vBulletin
Hope this is some help. There's a pretty good paddling community in Reno, so you should have no problem finding appropriate companions to head out with.
Bottom line--you'll have some great paddling out there too, it just won't be right in your backyard.