There are lots of fun things to do here, and June is a great time to come. One of our favorite things to do is go to the Gatewayy Mall- its a big, outdoor mall, and there are lots of things to see. There's a giant fountain commerating the Olympics, and it's set to music. During the summertime, there are tons of kids, (and adults too) that come to play in the fountain. The planetarium is also there, and has tons of free exhibits and things that are fun for toddlers and preschoolers to see and touch. There's also a small aquarium, with a touch tank, etc, that you can see in about an hour. The Children's museum is also supposed to be built there, but I don't know when it will be done, and I haven't been to the original one. (Although its pretty close to downtown too.)
Of course, if you're like most people coming to SL, you will want to go downtown and see Temple Square. You could probably spend all morning or afternoon there, and there are lots of places to eat close by. Temple square is beautiful- even if you don't want to take an official tour, its a great place for kids because the gardens are beautiful and there's lots of open space for kids to run around.
There's a museum in Lehi (about 25 minutes from downtown) called the Museum of Ancient Life (?) at Thanksgiving Point, that has a bunch of dinosaur themed exhibits. The Hogle Zoo is close to downtown and my kids love it- the admission is cheap, and the zoo is small enough to be managable in one morning or afternoon, but they also have all the great animals that you would expect to see. Plus, they have a great, great play area. Tracey Aviary is also a really good option for an outing- it's in Liberty Park downtown, and its a beautiful setting to go see lots of birds, including eagles, ducks, etc.
If you're up for a longer day trip, Park City is really fun. Plus, there are lots and lots of canyons to go to if you want to hike or just bbq outside. There are lots of kid friendly trails around depending on the canyon. You could also make a short drive up north and go see Antelop Island in the Great Salt Lake.
We had another discussion about this awhile back, you might want to do a search and see what you can find out. Let me know if I can be of any more help!
Violin teaching, doula-ing Mom to Abby, (8) Ashlynn, (6) : and Max (11/13/08) Diagnosed with Metopic Craniosynostosis. First surgery 5/1/09, Second surgery March 2010.