My introduction to the Museum of Peoples and Cultures was when I was a kid in home-school. My mom would take us to all the best educational exhibits in Utah Valley, and MPC was one of our regular trips. While the museum has activities for all ages, it is a great place to take kids. The kids have a chance to learn about ancient cultures while parents can keep track of them easily in the smaller exhibit floor. As far as what educational benefits the kids get from it, years later as a BYU student myself, I am now majoring in history. So much of that comes my mom taking me to the historical sites and museums around Utah.
Timpanogos Cave is located in the mountain (Timpanogos) to the North of Provo. The hike to the cave is fairly easy (up a paved path) and you can pay to tour the cave with a guide.
It is not open during the winter months.