If you plan on camping or staying at a lodge in the Park, book your stay at least 2 months in advance. Spots fill up quickly, especially in the summer. Download the Yellowstone app prior to driving to the park. It is very helpful, but there is little-to-no internet/wifi access in the park to use it. Be sure to pick up a Ranger’s wildlife sighting guide for your kids as soon as you get to the park.
Some areas of Yellowstone are not stroller accessible. Make sure to bring a backpack or sling for little ones. In general, keep close to those kiddos who tend to wander off. Signs everywhere in the Park indicate that the ground is unstable and visitors should keep to the paths. Be prepared to possibly sit in “traffic,” especially during peak season.
The earlier the start, the less people are out. As it gets closer to lunch, more people are out and about.
I personally would not take small children to Yellowstone in the winter—especially if you are camping. Even in August, the temperature dropped to the low/mid-40s during the night and in the morning. By noon it had warmed up considerably but I can imagine the winters in Yellowstone are brutal.
National Park entrance fees can be spendy, so if you plan on visiting more than one in a year, purchase a National Park Pass. Fuel up and grocery shop outside the park. Gas and groceries inside the park are more expensive.