I'll defer to others on something near Denver but Rocky Mountain NP is not far off the route so I'd start looking there.
The Colorado Plateau, as the unique area of Utah, SW Colorado, and northern Arizona and New Mexico are known, have a lifetime of hikes that you will want to go back for. For starters I suggest what is known as the "High Five", "the five premier hikes of the Colorado Plateau" as my t-shirt boasts. They are not that high and fairly easy to do.
1. Navajo Loop - Bryce Canyon NP There are a number of loops that can be combined with this to take in more of this fascinating geology.
2. Delicate Arch - Arches NP This is only a couple miles one way with modest elevation gain but it is almost entirely on south facing limestone which bakes in the summer sun. Bring plenty of water and sunscreen.
3. Angels Landing - Zion NP Start this early in the morning before the sun gets too strong because once the canyon approaching it is in the sun, it bakes, and it is steep. There are numerous switchbacks so that eases the grade of what is mostly like a paved path. The last half mile is what turns some people around but I don't think anyone who is comfortable peakbagging will have any problem here. You can get a separate t-shirt bragging about this one. I'd also suggest the Narrows in Zion, a hike up the Virgin River where you wade part of it. Be especially aware of weather outside the park because you can get surprised and trapped by a flash flood in here.
4. Cliff Palace - Mesa Verde NP There are several cliff dwellings you can visit in this park but this is the most notable one. You must hike it with a guide and they are regularly scheduled during the tourist season. The hairy part is the ladders you use to climb up into the chambers.
5. Rim Trail - Grand Canyon NP This is located on the south rim which is a zoo during the summer. It is about 13 miles long and offers spectacular vistas as well as the most notable attractions within the park, short of descending into the canyon. It is flat and I'm glad we did it in winter, when there are still some crowds but not as bad as summer ... the fact that there was some snow and ice on some exposed parts of the trail made it a bit exciting but the big thing here are the different views you get. Frankly, I'd defer this and stick with the North Rim. There are several good hikes including one in which you can drop into the canyon as far as you have time (and reservations) for. Also, the north rim being 1000 ft. higher than the south rim, it is cooler. It also is less crowded.
Looking at a map you might decide there is not enough time to visit Mesa Verde and then go to Moab so I sugggest you decide on either a northern or a southern route. The northern route would take you from Denver to Moab (Arches and Canyonlands), Capitol Reef is on the route and there are some interesting dayhikes there ... pick one ... then to Bryce Canyon, Zion, the North Rim and finally Las Vegas.
The southern route would take you to the Black Canyon of the Gunnison, Mesa Verde, Monument Valley, Bryce Canyon, Zion, the North Rim and on to Las Vegas.
If forced to choose, I'd take the northern route.
One more thing. Wear good hiking boots even though some of these trails are short and not very rough. The heat and hardness of limestone trails can wear on your feet.
