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colefeet I'm late to this party, but if you still have a couple of years here I'll give you some feedback based on areas where I've been and some of what I'd hope to do in that time frame...
Coyote Gulch - You can get in on the western end through the dry wash off the main dirt road or drive on further to 40 mile road and take that to the end. You then go down to a spot called "Crack-In-the-Wall" above the confluence of the Escalante River and Coyote Gulch. Here you can either send down you pack over the cliff on a long rope or unpack it and go back and forth between the crack carrying everything piece by piece until you have everything through to put back in your pack. I did the former going down and the latter on the way back up. My pack got scuffed up pretty good trying to roll it off the cliff.
Kings Peak - Just so you can say you bagged the highest peak in Utah. Multiple ways up to it, but the shortest and most popular route is Henry's Fork. Very pretty in the upper Henry Fork basin with ponds, springs, waterfalls, lakes, and flowers. I'll just add the rest of the High Uintas Wilderness area here as well. Lots of great looking trail guides here on BCP to serve as a reference. Stillwater Basins, Naturalist Basin, Notch Mountain area, Red Castle, and so on all look like great destinations.
Mount Timpanogos - Great nearby day or overnight hike rich with features such as waterfalls, wildlife (mountain goats & moose), meadows, alpine basins, lakes, snow fields, wildflowers, valley views, etc. It's no wonder this is probably the most popular hike along the Wasatch front. As such, if you don't like crowded trails, pick a day in the middle of the week. A prime time may be in the middle of the week just after college classes and school has begun in the late summer or early fall.
Arches NP - Yes, it's often crowded, especially in the Spring and Fall, but how can you ever visit or live in Utah for that long and not get your own picture of Delicate Arch? I suggest getting to that trailhead at dawn no matter the season, whether it is to beat the heavy crowds and the ensuing parking and restroom chaos in the spring and fall or to beat the heat in the summer. Lot of other great arches and areas to hike throughout the rest of the time there. Still on my to-do list is the Fiery Furnace. That requires advanced registration though and fills up real fast.
Bryce NP - Just about anytime of the year can be beautiful up there. With the elevation it is at, being at the top level of the Escalante Stair-Case, you don't have to contend with the same level of heat in the summer as you do in the lower deserts down south. Fairyland loop is a great day hike in the park boundary and Willis Creek is a short little slot canyon just below the park that I'd still like to do next time I'm down there. Being able to backpack the Under the Rim Trail might be nice too!
Capitol Reef NP - This is the place to go if you want much lesser crowds than you'll find at Arches, Bryce, or Zion. Lots of little rewarding day hikes such as Hickman Bridge, Cohab Canyon, Sulphur Creek Narrows, Grand Wash, Cassidy Arch, & Capitol Gorge. If you really want to push out and get away from people and you got a reliable vehicle, you can venture out to Muley Twist Canyon and get some great views of the Water Pocket Fold and Strike Valley.
Canyonlands NP - I've been to the Needles overlook & Newspaper Rock area, and most recently the Island in the Sky district where I was able to run out to the famously photographed False Kiva (not on maps – it's the kind of place you only share with those you really trust because it has a sacred feel to it, but you can still find routes and guides to it posted by others on the Internet). The park is so massive, all I can say is good luck picking a place to start. I've yet to get into the more remote Maze district. Some of the roadside features can obviously become quite crowded, but due to the vastness of the park, it's not hard to find some seclusion when you want it.
Calf Creek Falls - A moderately flat 3 mile trail just off the highway in the outer edge of Escalante National Monument that leads to beautiful slickrock canyon waterfall that plunges into a good sized pool that serves as a refreshingly cold escape from summer's heat. A few Indian ruins and petroglyphs dot the canyon walls along the way. It gets pretty crowded in the afternoons, so if you want good pics with nobody in the way, get up there really early in the day or go later in the evening and pack a flashlight for hiking back in the dark. There's an upper falls that's more remote that also has a nice looking swimming hole. You have to find the little parking spot right off the highway just above Boulder and then scramble down a good stretch of slickrock.
Wind Rivers (Wyoming) - I've yet to get really deep into their back country, but went to a scout camp a couple years in a row in my youth up there between Pinedale and Green River Lakes and was able to do a day hike a few miles up the New Fork Porcupine trail starting at New Fork Lake. The Winds were so spectacular from the outer parts there, I can't imagine how much more spectacular they'd be deep in the heart of them. I definitely plan to get deeper into them myself in the not too distant future. This is another legendary range where the possibilities look endless. Cirque of the Towers and/or the Highline trail there would be places I'd look at starting with based on what I've seen on BCP here.
Finally, here are a few additional areas within 4-6 hours drive or less from SLC that look worthwhile from my own bucket list that I've yet to visit:
Teton Crest Trail (Wyoming)
Wirepass-Buckskin Gulch-Paria Canyon (Utah-Arizona)
Grand Gulch & Bullet Canyon
Zion NP
Deseret Peak
San Rafael Swell, Little Wildhorse Canyon, etc.
Goblin Valley SP
Snow Canyon SP