Welcome to the RoadTrip America Forum!
What kind of budget are you looking at for this trip? If you truly are going to be out there for the majority of the summer, that's the better part of three months, or 12 weeks, and can ring up a substantial tab, even if you camp most of the time.
As far as camping goes, sometimes KOA is the only option in a particular area, but usually not. Their prices have gotten quite a bit out of hand in the past few years, so I would strongly recommend seeking alternates, such as state parks. Many of the state parks have significantly lower fees while still offering the same types of facilities that a private campground would offer.
There are multiple different states you can pass through on your trip from California to Colorado. I'm not a climber (more of a hiker), but I think Utah would give you the most options.
For those days when you will not be out climbing, what types of things do you enjoy?