It looks like all the suggestions so far, while great, are a couple hours from Salt Lake. One thing I love about Salt Lake is that there are fantastic hikes very close to the city. I would recommend going up either Big or Little Cottonwood Canyon. There are dozens of hikes in both canyons, and you can't go wrong with any of them. You can hike to peaks, lakes, waterfalls, etc. There are hikes that are great for kids (donut falls), or for a nice morning hike to beautiful aspen lakes (Lake Blanch, White Pine, Red Pine, Lake Mary, Lake Catherine, Twin Lakes, and more), or a more involved hike to a peak.
There are other canyons that are also pretty, but the Cottonwood canyons have higher elevations and are more alpine-ish. Anyways, let me know if you have any other questions. Hiking in Utah is one of my favorite things, and I really miss it as I am living out east.