With your timeline, I don't think that it's really possible to see both the Grand Canyon and Yellowstone - and certainly not have time to squeeze in a raft trip on top of it all. You could drive between those points, but you'd be so rushed that you really wouldn't be seeing anything at all.

The best case I see - you leave Vegas very early on the 18th and do a rushed 1/2 at GC. It's close to 2 full days of driving from there to Yellowstone, so that's the 19th and 20th. That puts you at the 21st and 22nd to see Yellowstone, but remember, it's then another half day drive to SLC and a full day drive to Denver. Even at a fairly optimistic/pushed pace, you're basically giving yourself the bare minimum amounts of time to see these places.

As much as I love Yellowstone, on your timeline, I think it would make a lot more sense to see the Grand Canyon, and then enjoy some of the other great National Parks between GC and SLC or Denver. You'd have about a dozen places you could pick from, including Zion, Bryce Canyon, Arches, Mesa Verde, perhaps even Rocky Mountain NP - and getting some whitewater in along the way too. Doing that would give you a lot more enjoying and exploring relative to covering miles. Alternatively, I guess you could probably make Yellowstone work as well, but then I'd skip the Grand Canyon, and that would really make it pretty important to fly out of SLC instead of Denver.