Most peope that know Colorado will tell you to go beyond Denver for the "true" Colorado experience. I've lived in Denver, Winter Park and now on the western side of the state. The longer I live here, the more I realize just how diverse the state is. Here are just a few of my opinions.
As already stated, the mountains around Ouray have got to be the most spectacular. However, its nowhere near as resort like as the ski towns. Great hot springs. Most of the recreation is centered around hiking and four-wheeling.
Someone mentioned Crested Butte. I love CB in both winter and summer. Great skiing, mountain biking, hiking and fishing. CB is also on the remote side, but has more amenities than Ouray due to the ski area. Its also not real easy to get to. On the up side, its not as crowded.
I'm partial to Winter Park/Grand County. I lived there for several years and still have family there. It's much easier to get to from Denver and no shortage of activities. Grand Lake is just as close to RMNP as Estes, but less crowded. Check out Devil's Thumb Ranch. Shameless plug, my brother works there. Quick Google should get you to their web site.
Grand Junction makes a nice hub for accessing the western side of the state. Its a fair sized city with easy 2-3 hour drives to a lot of great places (Aspen, Ouray, and even Moab,
UT another place everyone should see.
For something different, one of the best kept secrets of Colorado is the Great Sand Dunes National Park. An amazing place right where you would least expect to see it.
Sorry for the long winded response, but I could really go on. I'm very passionate about the state in which I reside. Please feel free to PM if you would like more detailed information or any help planning your trip. Enjoy Colorado, wherever you go.