All of the driving times are quite long in the park, since most formal attraction sites are 30 miles or more apart and traffic can be congested due to construction or animals. You should plan on being in the car often up to an hour when driving around but the scenery is great and there are places to pull off.
We've stayed at Fishing Bridge in our 13' Burro
and were swamped by the big rigs. We've also stayed at West Yellowstone, which is west of the western park entrance. There are several full service campgrounds there. We liked the KOA, but it's about 7 miles west of the town, adding to the drive.
In late May, you can certainly find snow somewhere in the park. Just be prepared and check ahead for road conditions. We've pulled both the 13' Burro
and a 16' UHaul
VT through the park and you'll see lots of rigs of all sizes. Also be aware that some of the park entrances may still be affected by heavy snow cover and be closed. The Park Service should have information on a daily basis for you.
As to a route, I imagine you would try to come down through Montana and enter at the north entrance. This was the old original entrance to the park and is beautiful! From there it is about 75 - 90 minutes to West Yellowstone, if you plan to stay there. Road speeds are not fast! When we stay, we try to cover a quadrant of the park each day over 4 days or so.
Don't wait too long to make a reservation. These places fill up fast!