re: Yellowstone/Grant Teton best places to stay

A couple things to know OP, May is mud season the snow is melting and things are muddy. Mid June is really the very beginning of their "summer". If you go before mid June you could very well run into snow and mud the same week. So be leery and understand that.

People think of southern summers where nice weather starts in April and its hot as hell by June, their only real summer in Yellowstone is July and August.

As said, you ideally want to move around. Everything is expensive because its remote. I would spend 3-4 days in Jackson Hole and then a couple days or more in Yellowstone.

You want to move around Yellowstone though because its huge and the road is basically in a figure 8 circle and there's no reason to backtrack really except for lodging. There's bison all over and traffic and mountain roads so its just slow going.

ETA: I'm bias because my FIL had a place there but I really like staying in Teton Village. Yes its a ski mountain, but the location is very good at the back gate of GTNP less than 1 mile away. You can take the Gondola up (check the calendar) to the top of the mountain to hike and enjoy the views and have a beer and appetizer when its open. Then as said its the back gate of GTNP.

I'd suggest 3 nights Teton Village to explore Jackson hole town, Teton Village, and south GTNP. Then drive north and see the northern part of the park and stay in Colter Bay or Jackson Lake Lodge. Jackson lake lodge has a great restaurant location.

Then hit up Yellowstone for 2-3 days working either clockwise or counter around the park slowly.

Bring binoculars

ETA; There's tons to do, you can easily spend 2-3x that amount of time.