If you'll have snowshoes and want to stay away from any avalanche danger, you can park at Solitude ski resort and cross the street and snowshoe to the ridge that separates Willow Fork from Mill F Fork (aka USA Bowl). If you stay on the ridge proper, you'll avoid any avalanche issues, and once you get above the trees, you get really nice views down Big Cottonwood Canyon. You might want to go all the way to the Park City ridgeline.
Another very popular snowshoe is to park at the "Spruces" Campground/Parking Lot halfway up Big Cottonwood. Cross the street and snowshoe into Mill D North Fork and go as far as you want. If you stay in the belly of Mill D North and stay on the most-beaten path, you'll avoid any avalanche issues. It's a a nice, mellow, wide canyon.
You can also do a similar snowshoe in Butler Fork in Big Cottonwood, closer to SLC than Spruces. Park on a plowed area off a curve.
Another option is to go to Alta and snowshoe east of there into what's called Michigan City, basically the northeast part of Patsy Marley mountain. If you stay on the snowcat track, you'll probably be safe from avalanches, but it's very easy to get into trouble in the larger area.
Just noticed you mentioned Olympus -- haven't been there in some time, but I'd bet you wouldn't need snowshoes or any snow gear right now. We've had a light season, and that aspect of Olympus gets a lot of sun. Someone else on here probably knows better, though.