Probably one of my favorite places in the world (that I've been to anyways). We have been twice. Once in the summer like end of June early July and once right when the parks opened end of April. Totally different experiences each time.
In the summer the crowds were just insane. We saw very little of Yellowstone in the summer because it seemed to have a lot more people than the Tetons and honestly I'm just not a horde of people person. Plus you couldn't get parking at any of the stops etc. Still what we did get to see was beautiful. We accidentally did a 6 mile hike trying to see a waterfall without having to take the ferry (because of course it was packed and no parking) and that was one of the highlights since not many people were on the trail. Keep in mind this was in the before times 2019 so probably even more packed now.
Anyways in the summer I honestly thought Tetons was 100x more enjoyable than Yellowstone. Less people, IMO more beautiful and we saw way more animals in the Tetons. However when we went in April I felt the opposite. First both parks were pretty much empty. Still snow on the ground and cold as fuck but totally worth it to have the parks to ourselves. We went with our moms and some older people so we weren't going to be doing any hikes or anything like that anyways. In April we enjoyed Yellowstone more and saw more animals there than at Tetons.
While the parks are next to each other they are pretty big, especially Yellowstone so I would recommend staying some time near each. The ideal setup IMO would be to fly into Jackson Hole (beautiful airport) and stay in Jackson Hole while visiting Tetons. Then go to Yellowstone and stay in West Yellowstone or near one of the other entrances when visiting Yellowstone. Or if you can stay at the Old Faithful Inn even better. They have other lodging inside the park also.
I would also second the recommendation of the bear sanctuary in West Yellowstone. Probably would not go out of my way, but if you stay in West Yellowstone it's worth checking out.
This was the patio area of the place we rented in West Yellowstone in April.
April in Yellowstone
Summer Tetons
Summer one of the inns or lodges in Tetons
