Great idea!

There are several ways north to Jackson. I would recommend first a stop at Antelope Island SP to visit the Great Salt Lake. You might be able to see some buffalo. There is an old farm house with docents.

You could take a side trip in Ogden to see the Eccles Dinosaur park. Lifesize dinosaurs with noise, very kid friendly.

Then you have a choice. You could continue north on I-15 to ID turning east on hwy 30 to Lava Hot Springs where you could stay the night. Fun town built around the hot springs. There is the large pool at the entrance to the town and there are the town hot springs in a park at the center of town.

Or you could turn east at Brigham City on Hwy 89. head to Logan then Logan Canyon and Bear Lake. Stop in Bear Lake for a raspberry shake, Stay the night in Garden City or Montpelier

Head to Montpelier and stop at the National Trail Center museum. I personally have never been able to see this, it is always closed when I am there, but I hear it is great for kids as well, it explains about the pioneers along the Oregon and California trails. (Mormon trail didn't go through Montpelier, it was further south).

Then follow 89 to Jackson, there is a cheese factory and store near Afton we always stop at.

Whether to take one or two days to get there really depends on when your plane arrives and whether you want to sightsee along the way. It is only about 5 hours to Jackson via 89

I know it is tempting to get to Yellowstone from Jackson, but it is a long drive. If you want to take one day try to head to Old Faithful only. You will have a taste of the geysers that way, but I wouldn't try to go anywhere else unless you get lodging in the park for one night. You would also see the lake at West Thumb. It is about 100 miles from Jackson to Old faithful. Plan on driving 25 to 45 in the park.

One of the funnest things I used to do with my son on the way to Yellowstone...if you go that way...there or back is stopping at the Bear World just south of west yellowstone. The bears are wandering about while you drive through the park very slowly. It's fun. Takes about an hour.

When I head to Jackson, I usually go through Idaho Falls, which is a city of about 45, 000, so there are lots of services available. From Idaho Falls, you can go through Swan Valley, ID and stop at a cute country store and get a square ice cream cone and then choose 1 of 2 routes into Jackson. You can go over the pass through Victor (where there is a dutch oven dinner theater) and Driggs or you can go along Palisades reservoir (which is a bit longer, but not as steep).

The Bear World mentioned by the other post is only about 20 minutes from Idaho Falls (its on the way to Rexburg and Yellowstone). I agree that trying to get from Jackson to Yellowstone is a long drive.