Suggestions on campgrounds for tent camping near Garden of the Gods or Seven Falls, Colorado Springs?
It depends. There are none super convenient to both. So, how far are you willing to go to get to seven falls and GOG? And what kind of amenities do you desire? (meaning both campground amenities and things like "a creek", "a lake", "in the mountains"). I used to live in the Springs and the campgrounds near Rampart Reservoir and a few in the mountains were my norm. I do know of a few others, some of which I have in this answer.
Lone Duck Reservoir in Green Mountain Falls is closer to Garden of the Gods. It is easy enough to head down US 24 to CO Springs to get to either of your desired destinations, but will take a bit of time (10 minutes to the back entrance of GOG and 1/2 hour+ to seven falls). And, from Seven Falls, you can head back up North Cheyenne Canon Park and down Gold Camp Road to get back to Manitou Springs. It is a gravel road, but has some great views and a few tunnels.
South of the Springs on 115 is Cheyenne Mountain State Park (near Fort Campbell). It has about a dozen tent spots. Not in the mountains, like Lone Duck, but also a bit more remote in some ways. Much closer to seven falls, but farther from GOG.
Rampart Reservoir has campgrounds on the reservoir. You can drive down Rampart Range Road to Garden of the Gods (dirt) and get some great views of CO Springs, or you can go through Woodland Park and then travel down US 24 into the Springs. I think these are really pretty and semi-convenient.
There are some on Colorado 67 north of Woodland Park that are nice, including the Red Rocks Campground, which has some neat formations nearby, but you are getting farther away. Beyond that you are getting quite some distance.
My favorite is Goose Creek Trailhead #612, but it is well into the mountains above Woodland Park and down a rather hairy dirt road.
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I think the best option would be Cheyenne Mountain. Seven Falls is located really close to the park. The park has really great hiking trails and waterfalls as well. http://cpw.state.co.us/placestogo/parks/CheyenneMountain/Pages/Camping.aspx