Utah Whitewater Club forum:
Unfortunately I just started last summer and haven't had time, flows, or friends to run much so my main winter hobbie has been reading about kayaking with a little bit of playboating. But I've researched a lot, and here's some basics, based on research rather than experience, that may get you started in the right direction.
Southern Utah is a desert, thats why no one probably replied. There just isn't much to run, and when it runs, its not for long. The Virgin will probably run, and has some awesome scenery and decent whitewater in spots, but i suspect it won't run for long this year (like it did last year) and may be in late April more than May; snowpack isn't great for that area and its already getting warm. Early may could be a good bet though, to do some stuff on the virgin; lots of options from II to V. To get you started: Zion Narrows would be worth running for the scenery alone, but its a long day or an overnighter, with a slow hike/drag in, and no escape once you're in. From 250 to some where around 500 the narrows is fast, scenic and fun class III, above that it gets more hairy, I hear. Below the narrows, there is class II - manky V roadside through the Park, and the Park runs the shuttle for you! Below that somewhere is the class V timpoweep canyon, and then there is reporedly some good stuff down below St. George in the Virgin River gorge too that doesn't often run. Flows can be different on different sections as there are various dams below the park.
Price River (more central Utah) from Woodside (near Green River, UT) to the confluence is reportedly really good, uber remote, and very scenic from about 400 to 700 cfs, but not many people really run this section. Apparently there is a lot of class 3 with one easily portagable 4+ that has some bad consequences. Makes for a really long day (24 miles or more) or a nice overnight and the shuttle isn't too bad at all. You take out on the Green River and can either take out immediately or combine it with a few miles of fun class II if you want. I'd like to do it this spring.
Cottonwood Creek (one of the hundreds of Cottonwood creeks in Utah
) below Joe's valley is a fun creeky III-IV roadside run that is popular and gets great reviews.
Left hand Fork of the Huntington, nearby, is a tight and technical IV+ creek that the locals love, when it runs, which isn't often or for long.
Black Boxes of San Rafael are classic desert V, when they run (once again not often or for long), but this is not a run to be taken lightly with multiple portages, seives, an unescapable box canyon, probable overnighter with hike in and out, and mega remote factor.
Try Nichols' guide to Utah Whitewater...
If you go to N. Utah there should be some fun IV stuff running on the Logan, Ogden, Weber, and Provo around then.
Once again, you'll have to talk to someone else for some real experience, but I am a fiend for scouring the net and books and learning as much as I can about the limited whitewater in my state and those are some that are on my list.