Starting about the end of Feb to the end of May is my favorite time of year for exploring Southern Utah, and we head down there multiple times during this time frame every year. While temps getting quite cool, even below freezing sometimes are common, daytime temps in the upper 50's to high 70's are quite common and great for hiking. You'll also find the crowds greatly reduced which is great. If you do get a snow storm, it generally doesn't last on the ground for long, and the place looks almost magical with snow.
Once you hit Boise, the odds of your hitting significant snow on the roads will go down a great deal, and if it does snow it tends to melt off quickly so if your travel schedule is flexible it shouldn't be an issue. Three Island Crossing state park east of Boise is just a little bit off the freeway, and will make a good overnight stop in an area likely to have reasonable weather mid March. Antelope island a little North of Salt Lake City might make a good stop for the next day. March/April are one of the best times to visit Antelope island because it's generally snow free and more importantly bug free. It's a good time of year to hike the trails and explore out there.
In Southern Utah you'll want to check on when the Easter Jeep Jamboree will be going as it's one of the biggest events of the year in the Moab area and will be quite crowded. Bryce Canyon will commonly get snow this time of year, but in my opinion that's a plus. For exploring there Kodachrome basin state park is a good option as they have full hookup sites available. Watch the weather if you'll be taking hwy 12 as the section over boulder mountain is pretty high elevation and might get snowy. They clear it off pretty quickly most of the time though.
Zion NP area is fantastic this time of year. Other than hiking canyons where you'll be in the water, you'll be there in some of the best hiking weather of the year. Most of the areas I like exploring in Zion involve quite a bit of hiking, which can be rough when the temps are well above 100 degrees in the summer.
As long as you have a decent heater and plumbing that can handle moderately freezing temps for a couple hours at night you should find it a great adventure.
