Tough question! All of the parks there are amazing in their own way. But since you can only do the 3, I'd pick these ones:
Zion: sits in a valley and offers some great hikes in the narrow canyons. If you hike there be sure to get all the info on the trail. Some can be dangerous due to flash floods. www.usatourist.com/english/places/zion www.nps.gov/zion
Bryce Canyon: The vistas of this canyon are amazing. The rugged pillars just don't seem like they should occur in nature. And hiking among them is magical! www.usatourist.com/english/places/utah/brycecanyon.html www.nps.gov/brca
Arches: The smooth rock of these arches are an interesting contrast to the other two parks. And the fact that the elements have created them over the millenia is mind boggling. www.usatourist.com/english/places/utah/arches.html www.nps.gov/arch
The individual entry fees for these 3 parks and the Grand Canyon add up to $85. So if you buy the Annual Pass at $80 you will save a little bit of money. http://www.nps.gov/fees_passes.htm And if you end up visiting an additional park, even better!
Highway 12 passes through Bryce Canyon. This is a very scenic drive. I'd recommend driving as much of this as you can. It certainly ranks as the most beautiful drive I've experienced in the U.S. http://www.scenicbyway12.com/\
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