Stretching your visit to 4 or 5 days has several advantages. You'll have more time to visit the roadside highlights, plus you can hike a few of the trails and actually learn from the exhibits at the visitor centers instead of just blasting by them.
The visitor centers are a good source of information about the volcanic activity, the flora and fauna, and the history of the park; knowledgeable rangers are on hand who can answer your questions. There are 10 centers and information stations, but some might be closed if you visit the park during the off seasons. The Albright Visitor Center is open year-round, and the Old Faithful Visitor Center is open during summer months and December to March. The others usually open concurrently with the lodging facilities in their area of the park. Ask the rangers about programs in their area, and check whether any geothermal features have undergone a change in their activity recently -- geysers have been known to revive suddenly after years of placidity.
The roads in Yellowstone are organized into a series of interconnecting loops, which you can come into from any of the park's five entrances. To simplify things, I'll discuss attractions and activities going clockwise along each section of the Grand Loop Road, beginning at the Madison Junction. But you can enter the loop at any point and pick up this tour as long as you are traveling clockwise. I haven't suggested an optimum amount of time to spend on each leg of the loop because that depends on your particular interests. If you want to get away from the crowds, take time to get off the pavement and onto the trails.
Watch Your Step! -- In thermal areas, the ground might be only a thin crust above boiling hot springs, and there's no way to guess where a safe path is. New hazards can bubble up overnight, and pools are acidic enough to burn through boots; so you must stay in designated walking areas.
