New Utah Museum of Natural History
I took my four-year old and two-year old. He loves all things dinosaur-related and she just loves hanging out with him, so they were quite happy to race around in the shadows of great dinosaur skeletons. While it was fun for them, I think it would be much more enjoyable for kids about eight years or older. There's so much written information, and things to be learned by interacting with the exhibits, that simply went over my kids' heads.
(If I had to choose one museum to go to with younger kids, I'd choose The Museum of Ancient Life which has more hands-on exhibits and the information is more digestible and relevant to younger minds.)
Some of our favorite things were walking across the clear floors and peering down at the bones beneath, curling up in a giant dinosaur footprint, and digging in the sand to excavate fossils. I would have loved to have spent more time at the exhibit that explained how they designed and built the new museum, but the kids were pulling me toward the huge skeletons. They are seeking LEED certification and so much thought went into it; I loved seeing the process laid out and explained.
Whichever museum you go to, Utah is definitely the place for Dinosaurs and amazing natural history!