A place I would recommend seeing on the way would be the Flaming gorge of the green river in the south. ... Seriously you should have a lot of the main sites relatively empty. Image Old Faithful without a crowd?
A terrific way of doing it -- especially if you can spend a week to ten days -- is to get off I-25 at Loveland, Colorado, head up the Big Thompson Canyon to Estes Park, go North through Rocky Mtn Nat'l Park, then into Wyoming through Flaming Gorge and then to Pinedale and Bondurant and from there to Grand Teton and Y'stone. Can't think of a finer tour any time of year.
H'ever, you can always find snow somewhere along the route in April. FWIW, we hit 13 below last night in Casper, and I don't think it's over, yet. And one other tip: Avoid Bob's Western Motel in (or near) Pinedale. Just take my word for it.