Hi Tom,
We've camped extensively in southern Utah. Kodachrome Basin State Park is a nice campground, but I probably wouldn't use it as a base camp if you're intending to visit Zion, Bryce and Capitol Reef. I'd probably pick something closer to Bryce instead. There is a commercial RV park just outside of Bryce (Ruby's Inn), or the campground inside Bryce is very nice, too. Which you choose would depend on the amenities you desire. Ruby's is a full-service resort and has a great pool, showers, restaurants, gift shop, summer rodeos, full hookups, and is a stop on the Bryce shuttle bus system. The Bryce Canyon campground doesn't have any of that (although they might have hookups, but we don't typically need them so have never inquired), but it's just a few minutes drive (maybe 10) to Ruby's. You can get campsites that are practically on the rim, or just a short walk. I'd look into reservations at either campground. One nice thing about the Bryce or Ruby's campgrounds as opposed to Zion or Capitol Reef and even Kodachrome Basin, is that they're quite high (~8000 ft!) so the summer nights will be cooler. It's probably about 1.5 hours to get to the east entrance of Zion, and two to three hours to get to Capitol Reef. The drive over Highway 12 through Escalante, Boulder, and to Capitol Reef is amazing. You drop into deep red rock canyons, see huge slickrock expanses and climb to almost 10,000 ft over Boulder Mountain. Here's a link for Ruby's:
http://www.rubysinn.com/
The Zion campground is also very nice. They have hookups available, and is very close to the town of Springdale. You can walk to the visitor's center where you catch a shuttle bus into Zion Canyon. It's lower in elevation at about 4000' and will be much warmer at night than compared to Bryce. It's kind of far for a day trip to Capitol Reef, though. Again, look into reservations. Here's a picture of one of our Zion campsites:
Tell me about the kinds of things you like to do (hike, bike, scenic drives, whatever), and I can make suggestions for you. We're hikers, so can offer the most info in that area.
You're going to have a blast. It's an incredible area!
Jeanette