A website that might be useful for you to get to know the basics about some of the parks, driving distances, etc, is:
American Southwest
A very popular week long trip is to fly into Vegas and to visit the North Rim of the Grand Canyon, Bryce Canyon, and Zion. I love Zion. The most popular hikes there are the Narrows (we did it as a backpack) and Angel's Landing, both of which I would highly recommend. I would also highly recommend a canyon like the Subway or Keyhole if you are a rock climber or if it is within your budget to go with a guide.
I also really like Escalante. It's one of the places we visited that we truly found solitude. Coyote Gulch (backpack or dayhike) is one of the most beautiful places I've been and I highly recommend it. The dayhike of Peekaboo, Spooky, and Brimstone gulches is a lot of fun (three dry slots that are fun to explore).
I also really enjoyed Buckskin Gulch in the Paria-Vermillion Cliffs area.
Moab also makes a nice base to visit Arches and Canyonlands. If you like to mountain bike, this is a great destination for that.
FYI, we have our pictures and trip notes online if you'd like to see:
Southwest Adventure '06 (includes Escalante, Kodachrome, Bryce, Zion, Paria-Vermillion Cliffs, Grand Canyon, Havasu Canyon)
Moab '03 (includes Arches, Canyonlands)