Shelf Road, near Canon City is a popular place to climb at that time of year. It is usually a good 10 degrees warmer than Denver, almost always sunny... Oh and jaw-droppingly gorgeous. Beautiful red limestone cliffs that go on seemingly forever. There is a campground, which might be full of rowdy climbers, but the surrounding area is all BLM, so you can camp pretty much wherever you want.
You can make easy day trips from there to the Royal Gorge, The Arkansas River, and the Sand Dunes NP.
Here's a page with more information (mostly about the climbing): http://www.mountainproject.com/v/shelf-road/105744267