Suggested time: 1 day
Arrive here, because flights to Vegas are cheap from almost everywhere, and since even hotels are subsidized by the casinos, you'll find lodging surprisingly cheap. Ride a roller coaster and have a drink.
If you're on a budget and want to stay for longer than 1 night.. there are various 4x4 trails about 45 minutes south of the strip where it is possible to camp.
Hoover dam should be interesting for many people who aren't interested in Sin City, and at any rate will be on the way out of town for many travelers.
Zion Nat'l Park
Suggested time: 1 or 2 days
Zion's main entrance can be a bit of a circus during peak season. Except for through-traffic, most people won't drive into the park. Instead you must park a few miles from the entrance (even this will be a problem), walk to a shuttle stop, and take the shuttle. Most tourists will want to ride the shuttle to one of the day-hike opportunities, Angel's Landing (some scrambling+exposure) being the most iconic and panoramic.
Views are incredible but crowds can be obnoxious in the summer, and all the nasty tourist-funnel effects are strong with non-backcountry Zion. Camping requires permits, parking is difficult, trails are noisy. Visitors expecting a peaceful setting or a private hike will probably be disappointed, and the crush of people and expensive lodging are enough to make me think it's best to get in and get out. For travelers not looking to continue immediately onwards to Bryce and planning on spending at least 2 days in Zion, visiting the Kolob Canyon area may be an attractive, less visited alternative to the main Zion valley circus.
Bryce Nat'l Park
Suggested time: 1 Day
Perhaps the most intensely unique attraction, the colorful hoodoos of Bryce Canyon should really be a must-see for anyone who hasn't experienced them; this is one of the most unique rock formations on the planet.
Bryce is incredibly impressive, but relatively small and accessible. If you're on a tight schedule, don't plan on spending more than a day here.. just drive around to as many separate overlooks as possible, do a hike if there's time, and then leave in the evening.
Serious photographers should plan to spend at least 2 days here and go on several hikes to shoot up at hoodoos; obviously overlooks will tend to offer only an elevated perspectives.
Grand Staircase-Escalante Nat'l Monument
Suggested time: 2-3 Days
A remote and relatively hostile environment, this large and mostly desolate area is not for the faint of heart but is home to lots of hidden beauty if you're up for the challenge. Slot canyons, stunningly beautiful rock formations, and access to back-country Glen Canyon are just some of the attractions. Stay on the edges of the Staircase if you can't carry extra water and gas, and high clearance 4x4 is strongly recommended. If DO you want to go into the interior, I suggest taking at least two days to drive the length of Hole-in-the-Rock Rd, visiting attractions on the way like Coyote Gulch, Zebra Slot, and Reflection Canyon. Many of the valleys in the Monument are sandy, but if you've got a car that climb the gnarly roads up onto the buttes then there are opportunities for mountain biking. The Staircase is huge, and distances between points of interest can be very significant. Given that, it's worth planning to spend at least a few days here.. especially since the area offers enough challenges to keep trekkers, climbers, canyoneers, riders, and drivers busy pretty much indefinitely.
If you're short on time, or just don't like the idea of taking your chances in some vast arid wilderness, then there are still options for you. Scenic byway #12 run adjacent/through Grand Staircase for part of it's length, and some attractions, like Paria ghost town, nearby Kodachrome Basin, and the Grosvenor Arch, will be accessible by smaller clearance vehicles.