A.C.
Hey there! I live in SLC. The great thing is it is so family friendly, so there are lots of cheap and free things for kids to do.
Some ideas: Living Planet Aquarium in Sandy. Sandy is on the south end of the valley, so it would not be a horrible drive from Provo. There is a stingray pool where you can feed and touch the stingrays, and you can see an octopus, sharks, turtles, all sort of tropical fish as well as fish native to Utah, there is also a new area with a cayman, tarantulas, snakes of all kinds.
If you are in downtown SLC, The Gateway is a great place for shopping and dining. It is an outdoor mall and there is often live music and a splash pad for the kids. While you are there, you could go to the Children's Museum which has so many fun and educational activities. You could also choose Clark Planetarium which has free admission (although you would pay if you went to a laser or light show). You could also tour the temple grounds which are beautiful.
In Ogden the Treehouse Museum is our favorite. It has so many fun things for kids to do. There is a fire engine with firefighter costumes, puppet shows, teepee, old schoolhouse, doctors office with bassinets and babydolls etc. It is a great place for imaginative play. They also do craft projects and reading and singing time.
In North Salt Lake (10 min north of downtown) there is a free splashpad and wonderful new park. Just get off freeway on 2600 S. and head West, go through Redwood Road and keep going a few blocks. The South Davis Rec Center is cheap admission as well and has great indoor and outdoor pools with an indoor waterslide and playsets built into the pools. It is in Bountiful about 15-20 min north of downtown SLC.
I know of 2 great parks- 1 in Kaysville which is between Ogden and Bountiful but you could stop at it as you are heading south, it is called Castle Park and it is on Nicholls Rd (right off Highway 89). It has play castles and drawbridges and is just a wonderful park in a beautiful area. There is a similar one in West Jordan which is in the salt lake valley on the way to Provo.
In Salt Lake City, our favorite park is Liberty Park, about 900 South and 500 West. It has a few playgrounds, tennis, a running track, a huge pond where you can rent paddle boats, feed the ducks and geese, also a splash pad, a swimming area (free), lots of trees and grass, an aviary (about $8.00 a person to get into aviary), a few carnival rides like a merry go round, airplane ride, swings and ferris wheel ($1.50 per ride). Definitely recommend.
Thanksgiving Point is near Provo, I have never visited but heard good things.
Lots of great places to hike as well. Park City has a huge farmers market on Sundays.
I would suggest you get on Salt Lake City groupon and keep your eye out for deals. For instance, I was able to get 4 tickets to Tracy Aviary for $20.00 rather than $8.00 each. It might give you some more ideas on fun things to do. Let me know if you have any questions.
**Not sure about star shows. I know I went to an actual star show there when I was a kid, but that was like 20 years ago. It looks like they have "light shows" which might have stars but I think they are like laser designs or something set to music. they also do educational movies there too.