Well, I'd avoid the I-84 corridor, which will take you along the Snake River Plain. "Plain" is a good way to desribe it--it's a lot of flat ground and sagebrush, intermixed with farms and feedlots. Not a drive I much enjoy.
After leaving Idaho Falls, you could consider taking Highway 20 west, which will take you near Craters of the Moon National Monument, which is pretty interesting if you're into geology. It could be on the hot side at that time of year, but it's kind of neat. From there you have a couple of options: you could go through Ketchum and Hailey (which is basically Sun Valley), or you could head north towards Salmon and Challis. At either point, you are on the edge of the Idaho Batholith, which is an ancient granitic formation that results in gobs of hot springs. ("Gobs" being a technical geological term, you know.)
If it were me, I'd go from Sun Valley through the Sawtooth National Recreation Area to Stanley. The Sawtooth range is a very beautiful place, and you'll be at higher elevations where the temperature is really pleasant. You'll get a true taste of Idaho up there. Then you could continue west through Lowman and Garden Valley (my hometown!), and there are Forest Service campgrounds along that highway with quite a few different hot springs. Kirkham campground is a very nice spot with good pools. Pine Flats is also good. From there, you can catch Highway 55 south into Boise and return along the I-84 corridor east, if that's the direction you need to go. Or you could back-track the way you came, or if you still have time, head north on 55 to Cascade and McCall, which are two more beautiful mountain towns with a different, less jagged style of mountains than the Sawtooths. Lots of hot springs around there too!
You'll find some good information on Idaho hot springs just through Google, so you can get specific temperatures and stuff, as well as directions there. I hope that helps--have a wonderful trip!