Base camp or travel in Colorado?

With 9 nights after your Colorado Springs visit, I'd pick 3 different places to base yourself out of and move around a little bit to experience some elevation and scenery changes.

If you want to hike in Rocky Mountain National Park, maybe stay in Estes Park at The Stanley (your teenagers might like that this is the place that inspired Stephen King's "The Shining"). If you want a place that is a little more hopping at night with more choices for hotels and restaurants, Boulder is only an hour away if you prefer to just day trip from there to various hiking spots.

There are many options!

Near Colorado Springs you can go to the top of Pikes Peak at 14,115 feet on a little train:
http://www.cograilway.com

And Royal Gorge is a short drive which my daughter and I did as a day trip from CSprings.
http://royalgorgebridge.com

Rocky Mountain National Park is north from Denver and you can stay in Estes Park, basically in the mountains at the edge of the park. There are many hiking and scenic drive options.

Colorado can easily keep you busy for an entire summer so you will have to prioritize.

Rafting on the Poudre River, Sand Dunes National Park, Summer concerts and chair lift rides in Vail or Breckenridge, Boulder and more are within a few hours drive of Denver.

I'm sure you know that public transportation is nearly non-existant except for downtown Denver.

Enjoy the planning and the trip!

If you enjoy hiking and nature, I would recommend Rocky Mountain National Park, Black Canyon of the Gunnison National Park, and Garden of the Gods. Another area of possibilities is around the ski resorts which includes hiking and mountain biking. Certain resorts have various things going on in summer, so I would check out places like Copper, Keystone, Breckenridge, Vail, etc and see if/what they have going on.

Things are spread out just enough in Colorado that it will be hard to stay somewhere centrally located. You'll have to move around, but if you plan your trip right, you should be able to camp out in a single area for a few days and then move on. For example, just stay longer in Colorado Springs if you wanted to see Garden of the Gods one day. You could then stay in Boulder which is relatively close to Denver and Rocky Mountain National Park. Boulder itself is a quirky little town that's worth exploring some. If you spend any time in the ski resort area, that's just far enough away from Denver/Boulder that I wouldn't want to commute back and forth (~2 hour drive from Denver). Just stay out there for a few days.

If you have traveled to other high altitude locations, you'll already know this: stay hydrated and don't overexert yourself on the first day. There's nothing worse than arriving in beautiful Colorado and getting hit with altitude sickness.

If you're willing to do some driving (I assume you're renting a car), you might consider a loop (in either direction) that includes stays in Colorado Springs, Durango, Grand Junction, and Boulder or Estes Park (as a base for Rocky Mountain National Park). The drives are very scenic. One day in Durango you should spend visiting Mesa Verde, which I suspect teenagers would be very impressed by, and another day you could ride the narrow-gauge railway to Silverton and back.

You'd have some long distances to cover, but with 11 nights, I don't think you'll feel rushed. It depends more on how willing you are to stay no more than 2-3 nights in a single location.

If you want to ride the Durango-Silverton railroad, better make reservations soon. Not sure if they still offer doing one way on the bus, but that would be my choice. While the scenery is great, a round trip on the train is almost too much slow travel! A friend from Colorado Springs once said to me that the San Juan mountains are the "class" mountains of Colorado. Distances are great from Denver, but if you can get to Durango/Ouray and/or Telluride, I don't think you'll be disappointed. I would personally choose this area over Rocky Mountain National Park. Black Canyon of the Gunnison National Park outside Montrose is a great stop. Note that creeks come from snow run-off and you will freeze your patooties off! Maybe you could consider the hot springs pools in Ouray or Glenwood Springs.

Thank you all for the great answers. Based on your responses, and other advice, I think the way to go is a road trip that goes in a loop and returns to Denver. Now the chore will be to figure out where to go, hitting great and meaningful places along the way, and preferably traveling along routes that are scenic in themselves, balanced against not driving so much that it becomes unpleasant. I guess the ideal trip would be to travel along some really beautiful route, stop at some interesting or beautiful activity along the way, continue on to some place to stay that is in a beautiful, stunning place, and then repeat. The opposite would be to travel along boring, flat or ugly routes, stop at an activity along the way that burns a lot of time to get to, is really expensive, and is crowded and disappointing, and then to go and stay at some lousy overpriced hotel that is not located on the side of a beautiful scenic mountain.

We live in CSprings, and while I agree with the other suggestions I would insert a personal comment here regarding the Durango RR. We did it this Spring, a day after we drove from Durango to Silverton, and we frankly enjoyed the drive more than the train ride. For us, the novelty of the train wore off after about 30 minutes. Unfortunately we were then stuck on the train for another several hours - it just became tedious after a while. Driving exposes you to the same (if not more spectacular) scenery with the option of stopping when you choose for a photo op or just to stretch your legs. As part of your loop drive you could simply do it heading north out of Durango along your way, and thus not waste time on an attraction which, for us at least, was an overrated and overpriced waste of a day. That said, Durango is a pretty little town and I do agree that using it as a base to access Mesa Verde is the way to go.
Might suggest orienting your route to enter Rocky Mountain NP from the western entrance, then driving over the gorgeous highway at 11,000 ft. into Estes. You won't regret it.