I think the easiest route is I-90 to Ellensburg, take I-82 to I-84, then catch I-15 south from Salt Lake City. You then take 89 and 12 to the park. That is the way we went last time, about a month later. The is one pass that is long and steep but we had no problem with our 35' and toad. Just took it easy and geared down.
Our favorite camp along the way is the state park at Provo.
On another trip, I ran into some snow crossing over to Moab in early April but it didn't cause any problems.