The Winds are where disillusioned Coloradoans go hiking... i plan to be there for the solar eclipse in 2017 (?).
Some Colorado options, since I know at least a little about that region:
First, I have to say it- the Chicago Basin in the Weminuche Wilderness. (I'm such a wit!) But seriously, the basin is stocked with 14ers, some of which are non-technical if you are so inclined. There's a big loop you can do. It can be accessed via the Durango & Silverton railroad, which is neat.
Heck, anywhere in the South San Juan or Weminuche wildernesses would be a classic alpine hike. Lots of options.
We have Rocky Mountain National Park. But it's still a national park- tourists, crowds, Xanterra, bear canisters, etc. And if I could only do one trip and wanted to do the national park thing there are other places I'd put ahead of it- Glacier, Grand Teton, etc. Yellowstone too, maybe, but I'd want to find a very remote hike to avoid the crowds.
Maroon Bells is also classically alpine and so beautiful it looks Photoshopped- but there can be crowds. Actually the crowds are only around the Bells themselves, specifically around the trailhead that the Aspen bus line runs to. In the back country there is usually solitude to be found. Heck, just the DRIVE over Independence Pass into Aspen is breathtaking. Great brew pubs there, too.
Those are the places I'd put at the top of my list. There are many other hikes here, of course, but for a flatlander who wants to do something "Rockies" that's what pops to mind. The Winds and Glacier are probably closer.