Spent a week in Yellowstone and a day in Grand Teton (a short drive from the south side). A week was more than enough — we drove both the Upper and Lower loops multiple times and saw a ton of wildlife in the evenings: 5 bear sightings, hundreds of bison, elk, and more.
The park has stellar visitor centers with museum exhibits on the wildlife and the science behind the supervolcano and thermophiles — worth stopping at.
July and August are the busiest months; late May and June are better times to visit. The park is huge — I'd recommend staying within the park if possible, and separating the loops by day.
Restaurants
West Yellowstone
In the Park
Canyon Fountain and Grill
Good for lunch
Old Faithful Inn Dining Room
Recommend reserving for the dinner buffet
Old Faithful Snow Lodge Obsidian Dining Room
No reservations needed
Upper Loop
Sightseeing
Lamar Valley
Saw a lot of bison here — requires a detour but worth it
Activities
Old West Dinner Cookout
Great food, ate in beautiful Pleasant Valley
Lower Loop
Sightseeing
Brink of Lower Falls
Grand Canyon of Yellowstone
Grand Prismatic Spring
One of the coolest spots in the park
Grand Prismatic Spring Overlook
Short uphill hike to the viewpoint
Old Faithful
Grand Prismatic was more impressive, but still worth seeing
Grand Teton
A short drive from Yellowstone's south entrance — well worth spending at least a day here.
Sightseeing
Jenny Lake
Scenic cruises, various hikes, and more
Teton Park Road
Great views of the Teton Range along the drive