We would like your help with the following. We will spend the weekend of June 15 in New York and will need to be in Calgary (Canada) around June 28. Between these dates we would like some 10 days in the mountains. The idea is to fly AMS - NYC - XXX - (YYY) - YYC - AMS (XXX and YYY being Denver, or Salt Lake City or Las Vegas or .....) which, it seems, can be done on Delta/KLM at reasonable fares.

We (couple, around 50) have never been to that part of the US except in Las Vegas (which we are not too eager to go back to, but LV could be the arrival/departure point). Main idea is to do a road trip and see and tour some of the national parks in this area. Some easy hikes on some days, and see the scenery and the landscapes. We don't mind driving, however obviously we don't want to spend the entire 10 days behind the steering wheel. Rental car + motels/hotels.

Questions:
1) what would you suggest in terms of national parks to see and tour? (suggested itinerary)
2) how crowded is June 16-27 in this part of the US? Would you recommend to reserve hotels in advance or will it be easy to secure accommodation on the spot?

Thanks in advance for any replies.