Wyoming and South Dakota

June.
Don't miss Devils Tower, on your way to or from Yellowstone.
and don't forget the Tetons while you're in Yellowstone area. Teton National Park has some great cabins & rustic lodges.

June will better and if you are from humid climate bring nasal saline.

Deadwood, Black Hills, Lake Sylvan, Badlands, Mt Rushmore.

Do the Needles Hwy in Custer St Pk I'd wait to June

"What not to miss?" depends on you, but going by way of I-90, a short side trip to Devils Tower will use up a 1/2 day, and if you're into history, a stop at Little Big Horn Battlefield near Hardin, Mt., but then you'd have to back track down to Cody.
As you've already heard, May will almost assuredly be too early, even early June can still have snow.

June, and don't miss Custer State Park.

June.

May can have snow.
June is better. Start early in the Black Hills.
Custer State Park has some very nice cabins. They have an excellent website. This is an outstanding state park, close to Mt. Rushmore, etc.
Take 16 west from Custer, SD (or I 90 west from Spearfish, SD) to Buffalo, WY, take 16 over the mountains and up to Cody. Spend a couple of days at least in Cody and see the Buffalo Bill Western Museum, very good, and if the roads are clear, drive the Chief Joseph scenic highway and enter Yellowstone from the NE gate at Cooke City, MT.
There are several hotels and a multitude of cabins in Yellowstone, but make your reservations as early as you can. If you like the log cabin style, Old Faithful is for you with the old lodge, the newer Snow Lodge or cabins.
Or, the Lake hotel, lodge or cabins near Fishing Bridge.
Or you can drive through the park to West Yellowstone and find more reasonable rates at the several motels. This will be the closest you can get to the park but be outside the park.

Mystic Hills Campground, nice cabins Pine area can also rent 4 wheelers and ride some trails right from the campground.

June.......May can still be cold


Just outside Deadwood
Easy/scenic 1 hour drive to Mt. Rushmore and only 15 miles to Sturgis
Fish 'N Fry Campground