We went last summer, the first week in June...the first week the season opened. Our kids were 6 and 3. Our 3 year old still remembers Yellowstone and since that visit the Bison has been her favorite animal. She sleeps with a stuffed Bison every night.
Yellowstone, hands down, has been the best trip we have ever taken. We saw incredible, amazing things!
We heard from many people that the Grand Tetons are also nice. We only had a week, we wanted to spend it seeing as much of Yellowstone as we could. We did hear from many park rangers that that's the place to go if you want to see a Moose. The Moose is the only animal we wanted to see that elluded us. We were lucky to see Grizzly, black bear, elk, fox, wolves,bison,marmots, nesting owls....but we never saw a moose.
The park is most likely booked. When we made our reservations, I was told that most people book up to a year in advance. We stayed on the park. You may call and ask about waiting lists or cancellations. The park is so large, driving in and out daily would be tiresome.
The Frommer's guide to Yellowstone and the Grand Tetons is excellent. I usually don't read guide books, but this one really helped us plan our trip. Things we would have skipped (like the cowboy cook out) we did because they were recommended for kids, and I am so glad we did! This book also helped prepare us for the weather. I was expecting cooler than Texas, I was not expecting snow storms in June! We were prepared thanks to the book.
We each picked the one thing we wanted to see. We watched videos about Yellowstone on Netflix with the kids, so they could choose too. Be careful with the videos, some can be a bit scary for kids. You might want to preview them first. This helped us plot our travels. We did not over book ourselves. We saw one or two major things each day, so that we would have plenty of time for side trips, breaks, meals, etc.
Our kids both did great in the car. There were so many interesting things to see, we never heard complaints. It also gave them time to rest up or nap. We did make sure to bring plenty of snacks and drinks, all the hiking made them hungry! We also invested in good hiking shoes. I cringed at buying $50 shoes for the kids, but it was money well spent (and on a side note, one year later, the kids are STILL wearing those shoes).
If I had known there were so many board walks (especially at Old Faithful), I would have brought the stroller. We made it work. She rode on Daddy's shoulders when she got tired and my hubby didn't seem to mind. But had I known that a stroller would have worked in so many places, I would have packed it.
We didn't have any problems with the fumes. Most are blocked so you can't get right on top of them. So there were no problems with that at all. Well, the kids did make a lot of "rotten egg" and "fart" jokes, but I guess that is to be expected. :)
Food, gas, souvenirs are pricey! Get what you can outside the park. We bought t shirts (Lots of sales on those), but we ordered toy souvenirs on amazon before we left and hid them in the luggage. Glad we did that and saved some cash. Pack a cooler if you can.
Make sure to bring plenty of memory cards for the camera. You'll take more photos than you ever thought possible.
The cowboy dinner will be worth it for your kiddo. It's pricey and the food is just ok, but the experience is great. Our son also did the junior ranger program to earn badges. It's free. They had something for our daughter to do as well. The ranger swore her in and gave her pencils and stickers instead of a badge. And if you also have older children the young scientist program at Old Faithful is fantastic. You have to pay $5 or so, but it is so worth it! They give the kids laser thermometers to see how hot the geysers get. It made things really interesting. Let the kids talk to the rangers. They were all really friendly and taught our kids a lot. The ranger talks that are scheduled daily are worth the 10 minutes.
We flew in to Billings. I think the drive was a little under an hour. That drive is tedious. We let the kids watch movies. Once we made it to the park, we banned all electronics for the week.
I hope you can make this happen. It really was an incredible experience for our whole family. Feel free to pm me if you have any questions I might be able to answer. I am absolutely no expert, but I might be able to help.