For your consideration-
Bear Lake, Utah. It is on the Utah-Wyoming-Idaho border. You can stay at Ideal Beach resort (condos with full kitchens) right on the lake, which BTW is the most gorgeous turquoise lake nestled in the mountains, and a family suite for half a week is $1,208 + tax or there are motel/hotel rooms or even camping at Blue Water Beach next door for much cheaper. We go every year, since we live in Utah and have a family cabin up there, but there is seriously soooo much to do. At the resort, there are bbq grills everywhere, playgrounds, sand, horseshoe pits, volleyball, bikes for rent, those family bike cart things, tennis, pools and hot tubs, paddleboards, jet skis, waverunners and boats for rent, lots of yummy food in town (famous for their raspberry shakes) and of course swimming in the lake . Oh, there is also a golf course right across the street and mini golf at the resort. It is like heaven on earth, I think. If you fly for free and then just need a car rental (it is a 2.5 hour drive from SLC airport) then it should be in your budget. Also there is Minnetonka cave within a 30 minute drive- very neat cave- and Logan Canyon is one of the most beautiful drives in the world and there are lots of cool hikes in there- check out the Wind Caves hike, also Ricks Spring. Lots of fishing to be done in the Logan River or at the lake.
We also really like Island Park, Idaho. It is near Yellowstone- about a 30 min drive. There are lots of great cabins for rent- check out Pines at Island Park, which are super nice with full kitchens and hot tubs- or camp (Buffalo Campground is quite nice). When we go, we float down the Buffalo River all day- you can take your own floaties or you can rent canoes etc at Mack's Inn nearby. Mesa Falls is nearby- very lovely- also there are lots of lakes nearby for boating and fishing, and you can take a few day trips to Yellowstone which is incredible as well. The best part is you will see some incredible wildlife- last year we floated past a baby moose, saw blue herons, elk, deer, buffalo, etc.