US 89 through Livingston is fine, but once you get to Gardiner, at the north entrance to the park, you have two sets of switchbacks to get to most of the park - one part from Gardiner to Mammoth, the other part from Mammoth to the main level of the Yellowstone plateau. Slow going, lots of tourist traffic - then, unless you camp in Gardiner or at Mammoth (no hookups), you have a long way to go to get to West Yellowstone, with traffic, animal jams, etc. I'd recommend going along the west side of the park to West Yellowstone first. Haven't ever driven 287 so can't offer an opinion there; 191 seemed ok to us (northbound) when towing, but that was late September. As far as US 20 goes, the previous description of it is right on. Good road, lovely scenery once you get north of Ashton.
There has been a lot of snow in the park over this winter, so there could potentially still be some along the roadsides and even in some CGs. Fishing Bridge is the only CG inside the park with hookups, and reservations have been open for quite some time now. If you can get in, it's the best location, but we usually use Grizzly RV Park in West. FHU, laundry, clean & neat, and as someone pointed out, close to the west gate of the park and close to town. The KOA is about 8 miles west of town, and at least in past years has been more expensive than Grizzly.
May - especially after mid-May - is a good time to go. Some of the roads in the park may not be open (Dunraven Pass & Beartooth HWy) but the rest will. It can be chilly, so dress appropriately. Lots of new bison & elk calves around. Your time frame is about right (4-5 days); definitely spend one of them in the Tetons.