Coming in for the holiday weekend and we've never spent any time in the SLC area, though I've spent time in southern Utah. With access to a good corporate rate we plan on staying downtown but are open to staying elsewhere if needed. So, with so many options I'm putting a post out for advice.
Few things:
-Like to keep the trail heads within an hour or two of the city.
-Wanted to do Kings but I don't think it's a quick overnighter right now and I'm not bringing full camping gear. Tell me if I'm wrong here. I can bring a tent and a bag if things are good and we'd consider doing this as a long day if things were really good.
-Can bring ax and crampons if needed and we're comfortable with intermediate snow climbs and 3/4 class scrambling.
-We have a few days so a combination of short, long and maybe brutal are all in play. Most likely scenario will be a good 15 mile + day, tourist crap day and then another half day outing.

Thanks,

If you want to use your ice axe, here are some of my favorites:

Timpanogos
Pfeifferhorn
Lone Peak
Broads Fork Twin Peaks
Superior/Monte Christo (it's good from Lake Blanche too)
Box Elder Peak

Gobblers Knob or Raymond are easier than the above.

Shorter and easier than that are Grandeur Peak or Mt Aire, though the road to Mount Aire won't be open yet.

For snow free, try Antelope Island or Stansbury Island. Mount Olympus is a good one too. Storm Mountain form Ferguson Canyon too.

The north face of Olympus has some good couloir routes, such as the North Face Couloir or Apollo.

A little farther from SLC is Ben Lomond, Lewis Peak, and Thurston Peak to the north. South of SLC is Rock Canyon which is scenic.

PS, I would recommend staying elsewhere than downtown. Downtown isn't bad, but I much prefer places like Cottonwood Heights or Draper, which have an even better mountain backdrop.

Kings Peak would be a ski (or snowshoe) trip at this point.

Local peaks still have a lot of snow but would be doable. Pfeifferhorn is a good hike. Broads Fork Twin Peaks would be somewhat harder, but still easily done in a day.