I have lived in Utah all my life and am feeling the same way. Last year we went camping in the Uintas and it was awesome. Definitely one of those "hidden wonders". There are so many lakes up there, tons of hiking and fishing. It was a very interesting place because the mountains very much resemble Southern Utah but then there was also so much vegetation. It was a neat mix.
Another great place is Logan canyon. There is a 2 mile hike to the Wind Caves which is incredible. The path is quite narrow, so I would get a babysitter or take the kids when they are older. We went a few years ago and had a picnic on the top of the caves which overlook the whole canyon. Incredible. Last year I took my kids up the canyon several times just to picnic. The dams at the mouth of the canyon are a fun place to picnic as well. That canyon is just beautiful.
Bear Lake. I can't imagine a more beautiful lake! And so many kid-friendly activities. It is really inexpensive to camp at Blue Water beach or just pay a few bucks for day access. On the North West side of the lake are MInnetonka Caves, very interesting, although my kids were worn out with all the walking so maybe save it for a few years.
Millcreek Canyon in Salt Lake City has a ton of trails. We hiked to Dog Lake last year. I can't wait to explore that canyon more this summer.
We are taking our kids to the Salt Lake City temple grounds in a few weeks. I think if you live in Utah, you should be familiar with its history even if you are not LDS. It is supposed to be a very interesting and beautiful tour. You can also head up Emigration Canyon and visit the "This is the Place" park which focuses a lot on LDS history, also Hogle Zoo is right there as well.
In Idaho- definitely visit Island Park. What a great place to camp. We like Willow Flat campground which is up Cub River Canyon right out of Franklin, not too far of a drive for you.
If I think of anymore, I will let you know.