I am truly surprised, as a late arrival to this thread, to read how many of you posted a place that immediately came to my mind as well - Mesa Verde. I went there first when I was 9 years old, and I remained fascinated with it all my life. Since I live in the southwest, I have been lucky enough to go there quite a few times, and I never tire of imagining the original residents there, making their lives function through their ingenuity. Quite a fascinating place.
My second favorite is the Grand Canyon. Every time you go there it looks different, due to the way the light hits it. Growing up only 90 miles away, we went several times a year. As a teenager, I hiked to the bottom several times, camping out under the stars at either Indian Gardens or Phantom Ranch.
Another place that really impressed me with its beauty, and I've only been there once, was the Wisconsin Dells.
I was 14 years old before I first saw the ocean, and that awesome sight, at Morro Bay California, stays with me still.
Other favorites are:
San Francisco, CA
Rocky Mtn National Park, CO
Ozarks area of Missouri