Welcome to Oregon (soon!)! I moved from Utah to Oregon about 3 months ago. :)
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Are you going to be staying on 80 (through Park City) to go down through Salt Lake City before getting on I-15 (adding an extra hour or so to your trip), or will you be taking I-84 through Morgan canyon before getting on I-15 in Ogden? If you will be taking the SLC route, Park City has some fun options: the Utah Olympic Park (rides like zipline, alpine slide, bobsled - be aware that the rides are only open on weekend afternoons in September and not at all after that; tours; a museum; shows), the historic jail on Main Street (along with some fun ice cream/candy shops), great hiking (especially in early fall when the leaves are changing color). In Salt Lake, This Is The Place Heritage Park is a great place to visit. There are historic buildings and tours (homes, stores, schoolhouses, barns), and there are also sometimes demonstrations on farm living before the turn of the century. Also train rides, pony rides, petting zoo... Evenings near Halloween they do some kind of special "Haunted Village" thing. Temple Square in downtown SLC is a great place to tour and learn about the history of the area and about the predominant religion (I am LDS (Mormon) but it is a great tour for those who aren't as well). At certain times you can watch practices/performances of the Mormon Tabernacle Choir, organ recitals, and other performances in the Tabernacle. A free movie about Joseph Smith and the history of the Church of Jesus Christ of Latter-day Saints. Or just walk the beautiful gardened grounds. On the University of Utah campus there are several museums that would be interesting to kids the ages of yours- I think there's an art museum. but I know for sure there is The Utah Museum of Natural History. Check the website; they often have free days. The Clark Planetarium at the Gateway mall would be good, too. There is a small (free) museum area and then they do star shows and laser shows as well. There's a zoo and an aquarium in Salt Lake (zoo on the eastern bench near This is the Place Heritage Park, and the aquarium in South Jordan/Sandy area), but other cities around the country definitely have better... West of Salt Lake on I-80 are the Bonneville Salt Flats, which are pretty impressive the first time you see them. The Great Salt Lake is west of Salt Lake City; the best area for access is north of SLC in Layton: Antelope Island. You can learn about the ecosystem there and might see the resident wild bison and antelope and lots of migrating birds. For a real teaching experience (albeit a gross one), take your kids for a swim in the lake! Tons of tiny brine shrimp (sea monkeys) and lots of brine flies on the shore to eat the shrimp. And enough salt to let them float (and it is usually very shallow so the water is typically warmer than you would expect, and there are sandbars they can wade to as well, if they want to brave the flies and shrimp...). I think it is definitely something *everyone* should do at least once in their life! The island is also great for hiking and bike riding if it's not too hot. South west of the city there is a huge copper mine. I haven't ever toured it but it seems like something that might be interesting to older kids. It's called Kennecott Copper Mine.
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North of SLC, along with the Great Salt Lake, there is a bird refuge in Layton where you can go birding (there is also a bird refuge north of Ogden, near Brigham City. It is a bit nicer than the Layton one; it has a building with little exhibits and a boardwalk outside) . In Roy (just south of Ogden) there is an aerospace museum near the Air Force Base (Hill Aerospace Museum). It is free and there is a huge display of jets, along with a small "hands-on" educational room where kids can ride flight simulators, try to navigate a model airplane, learn about gravity and flight, try on flight/space uniforms, etc. Call ahead to make sure they will be open when you visit; the education center has weird hours... There are some great, great hikes all along the Wasatch Front. We lived near Ogden so I am most familiar with those, but some of our old favorites are Adams Canyon (east Layton), Waterfall canyon (east Ogden), Bonneville shoreline (all along the Wasatch Mountains on the east, in both Salt Lake and Ogden, and in between), Wheeler Creek (best access in Weber Canyon - 12th St. in Ogden). In my opinion, the best things to see in Utah are all outdoors, but are not all easily accessible from I-80 (National Parks like Arches, Zion, Escalante, etc, and state recreation areas like Capitol Reef, the high Uintahs, etc).
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Along the way from Utah to Oregon (assuming you are taking I-84), Boise has a few fun things to do (zoo, river, rose gardens, museums), and then somewhere near the Idaho/Oregon border there is an Oregon Trail Interpretive Center. We haven't stopped there yet (it's something like 8 miles off the highway each way, and we've always been in too much of a hurry), but it I were taking my time and trying to make it a "homeschool" activity, I would definitely visit. Closer to Portland, the Bonneville Locks and Dam is a great place to stop and learn about the river, salmon, and the dam. My kids wanted to watch the salmon swimming up the ladder for hours when we stopped! And then take historic highway 30 to pass by dozens of waterfalls, and stop at Multnomah Falls, and hike up at least partway, It's beautiful (and popular, so try not to go on a Saturday or Sunday!).
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Have fun on your journey! As a kid we drove I-80 quite frequently, from the midwest to Utah or California, and I still remember most of the places we stopped along the way. My mom would stop at basically any historical marker or view point. It made for long trips, but we saw some amazing things we would have never seen or learned about otherwise!
