C.L.
We went in August with our boys who are 12 and 15. It was beautiful, but be prepared to spend a lot of time in the car. Although they loved the geysers, mudpots and wildlife on the road they did get a little bored with all of the driving. Be sure to abide by all of the warnings and cautions re wildlife. Not to scare you, but there are incidents involving humans and wildlife there every summer. Food is only available at about four locations in the whole park and they all pretty much have the same menu except at Old Faithful, which has a larger cafeteria. It is also expensive. If you want more flexibility and want to save some money be prepared to bring food, snacks, water, etc. with you. Some of the hikes to see the attractions can be a little long for younger children, but we saw A LOT of very elderly people who could barely walk so there are alternate ways to view things. One of the highlights for us was the "boiling river," but your kids might be a little young for that.
You will need to decide if you want to stay inside or outside of the park. If you want to stay inside you need to make reservations now. Are you camping or do you need a hotel? We personally found it cheaper and more family-friendly to stay just outside the park. We stayed in West Yellowstone, MT near the west entrance of the park. There are four entrances and all of them have some sort of lodging somewhat nearby, although I think West Yellowstone might have the most options. Keep in mind that there is lodging in Cody, WY, but that is a one hour drive just to get to the entrance to the park. West Yellowstone is not really a tourist attraction in and of itself, but the big perk was that our hotel was 1/2 mile from the entrance to the park. The food in West Yellowstone was better than I was led to believe it was going to be. There is an IMAX theater in West Yellowstone and we really enjoyed the movie about the big fire in the park in 1988 (evidence of which remains in the park today). There is also a Grizzly wildlife center that is kind of touristy, but it was a safe and easy way to get up close to real grizzlies.
PM me if you have specific questions. We also spent a few nights in Jackson Hole, WY and drove and hiked through the Grand Tetons.