Red Canyon Park (not to be confused with Red Rock Canyon Open Space in Colorado Springs) is a 600 acre park located about 10 minutes north of Cañon City with similar rock formations as found in Garden of the Gods. It's a fantastic area to hike and explore.
To get there head north out of town on Field Ave, eventually it will intersect with Red Canyon Road (also known as Garden Park Road), turn right and drive about 7 miles further and the park will be on your left. Look for the gate you see above, it sits back off the road. For a map showing directions to the park, click here. For a map of the park itself, click here.
I'd highly recommend a high clearance vehicle with four-wheel drive for this one. The road is not well maintained and has many deep ruts in it. I wasn't too sure where to go when I first arrived, so I initially parked near a picnic pavilion and bathrooms, and started in on foot. Following the creek bed (which was mostly dry the day I visited), I encountered many interesting rock formations and narrow canyons (top two photos, left).
After returning to my car, I continued to drive on further up the entrance road. I soon came to views of much larger sandstone formations (bottom two photos, left). These are actually on private property, but can be viewed from the road. See the comment below from the landowner. Just to be clear: These formations are not part of the actual park. Respect private property, don't trespass, and just enjoy the views from the road.
Continuing on up the road, you'll find other rock formations that are within the park and accessible on foot. I haven't found any kind of trail map for the park, if you know of one, please put the link in the comments below. There's so much potential at Red Canyon Park -- but I think the condition of the road and the lack of defined trails really are a limitation. Still, if you have a high clearance vehicle, it's a fun and beautiful area to explore.
Check out the Gold Belt Scenic Byway website for a great guide to the roads leading to Cripple Creek from the Royal Gorge area.