We spent almost all summer in Colorado visiting Aspen, Vail, Snowmass, Breckendridge, Frisco, Colorado Springs, Keystone,... Even though every one of these places is special and unique my recommendation is to visit Vail first. It's not only great for solo travelers, young adventurers but also for families with kids as well. At 10000 ft, where the scenic gondolas will take you, are wonderful trails for beginners and experienced hikers too. The pine trees and flowers smell fantastic... There are even more bike trails there. Vail is a very sweet place for kids to fully enjoy many summer activities. As all mountain destinations, Vail has unpredictable weather too, so the best way to enjoy it is to start hiking/biking or other activity you like early and finish till noon. Then having a great dinner or shopping at the many really nice places would be a wonderfully spent afternoon. Nearby towns like Avon, only a few minutes drive from Vail, may have lower in price accommodations. Vail's free parking is very large and there is no problem finding a parking space. Not having a car is also not a problem at all, because there is a bus every 30 minutes from almost every nearby town to Vail and to the gondolas. Between the two gondolas there is also a free bus or you can just take a walk in only 20 minutes. Summer can be cooler if you choose mountains next time. Choose Vail for an extraordinary one.
Note: Vail is at 8022 ft so as everyone else recommends, I would also recommend spending a few days at lower elevation. Choosing Denver or Boulder at approximately 5000 ft would be a great decision. Take the time.