If you can spend Tuesday night in West Yellowstone that would be great. So then that would be Sat/Sun nights in Gardiner, then Mon/Tues in West Yellowstone and your remaining night in Jackson, right? I am kind of with PeachBelle, you could change that 2nd Gardiner night to West Yellowstone and make it 1 Gardiner and 3 West.
So here's what I would do:
Saturday afternoon: After your rafting, I would head in to the park. Go to the Visitor Center at Mammoth. If you want to get your younger kid started on a Junior Ranger program that would be the place to start. Tour the lower terraces, then drive the Upper Terrace if you have time. You might want to pick up the brochure on the walking tour of Ft. Yellowstone to do either this afternoon instead of the Upper Terraces or Sunday some time. Overnight in Gardiner
Sunday: Finish up Mammoth. Drive east toward Tower Jct., taking the Blacktail Plateau Dr., then take the short road back to Petrified Tree. There are often black bears in this area. There are a couple of waterfalls before Blacktail Plateau. Head on out in to Lamar Valley to see if any wildlife is out. Retrace you steps, then turn left at Tower Jct and go to Tower Fall, stopping at the overlooks if you can (there is road construction along here somewhere). The overlooks will be on the wrong side of the road, sorry! Head up over Dunraven Pass, stopping as you wish and keeping an eye out for wildlife. If your guys want a hike, you could do the Mt. Washburn, but it's about 3 hours, I think. At Canyon, make a pit stop at the Visitor Center, then head for Norris and back toward Mammoth/Gardiner. If you want a restaurant lunch today, you should probably stop at Roosevelt Lodge. There is not much at Tower...all I remember is hot dogs and ice cream. Overnight in Gardiner or West Yellowstone.
Monday: Head back south either over Dunraven Pass or toward Norris if you stayed in Gardiner. Your goal is Canyon. Stop at the VC and look at the exhibits. Then head south into Hayden Valley stopping if you see people with long camera lenses or spotting scopes. Do Mud Volcano, then Fishing Bridge. Get out and walk across looking for wildlife. This is a place where I've seen otters, eagles, white pelicans, muskrat. Continue East to Fishing Bridge General store if you want, if it's lunch time, this would be a good place. Your other choice would be Lake area which is just south of Fishing Bridge. Head back north thru Hayden again, then pick up the South rim of the Grand Canyon, then the North Rim. After that, head toward Norris. Do some of Norris if you still have the energy. then out to West Yellowstone for the overnight.
Tuesday: Make this your geyser day. Enter the park, turn left at Madison and head to Old Faithful. After you cross the bridge over the Gibbon River you will take the right turn into Firehole Canyon. Rejoin the main road, you will pass several geyser basins, but I recommend you skip these for the time being. You can pick them up on your way back to West Yellowstone tonight. So Go directly to the Old Faithful Visitor Center. Check on eruption times for the predictable geysers, then plan your walk thru the geyser basin. You can write the times down on the back of the brochure available at the desk (50 cent donation)...they have pencils too. For lunch here, I like the grill at the Lower General Store. If you are standing in the front parking lot (where the buses are parked) of the Old Faithful Inn with Old Faithful being on your left up the hill, then the Lower General Store is on your right. Try to time it so you also have room for ice cream at the Old Faithful Inn. (This is a must, lol!!) You get it on the 1st floor, then take it up to the 2nd floor deck overlooking the geyser basin. On your way back to West Yellowstone, stop at Biscuit, Midway, Firehole Lake Rd and Fountain Paint Pots, depending on how much energy your group still has. There will be a lot of walking in the geyser basin. They might like to hike up to Observation Point, which is above Old Faithful.
Wed: Head to Grand Tetons. You can go either via Canyon or Old Faithful. Backtracking in Yellowstone is never bad! It just increases your possibility for seeing wildlife! If you didn't work Norris in, you could do Artist Paint Pots and some of Norris this morning as well.
There is often a lot of wildlife on the West entrance road between West Yellowstone and Madison Jct. Have the guys watch the river for trumpeter swan and the trees for bald eagle. You should also see bison and elk.
Coming in from Gardiner, start looking to the left as soon as you cross the river. You may be able to see bighorn sheep up on the cliffs on the left side of the road. There are a couple of pullouts on the right, so dive in there and scan the cliff side.
Will you have some binoculars?