Hey, that sounds a bit like our trip back in 2008! At least the Yellowstone part - we didn't go to the Tetons. We flew into Bozeman, spent two nights at Mammoth, then two nights at the Yellowstone Lake Hotel, then three nights at Old Faithful Inn, then drove back to Bozeman.
From Mammoth Hot Springs - you definitely want to wander around the terraces that are there, and also go to the Visitor Center.
You might also want to drive to the Norris Geyser Basin (an hour or so away) - we enjoyed wandering around and seeing the colorful pools. Artist Paint Pots is not far from there but I was disappointed in that, and didn't find it to be very colorful.
From Canyon (or on the way to Canyon) you might want to go further east and visit the Lamar Valley. Especially if you go early you might see lots of wildlife - big herds of buffalo, maybe some antelope, and maybe even some wolves. Back on the road towards Canyon, be sure to visit Tower Falls. Unless they've managed to fix the trail the best views are pretty much near the top, so there's no need to hike down if you don't want to.
In the Canyon area are also upper and lower Yellowstone Falls, as well as two spectacular viewing points: Artist Point and Inspiration Point. You can drive to all of those, park, and then take a short walk to the actual viewing areas. If you're feeling ambitious and strong, you can take Uncle Tom's Trail down, down, and down to a place where you get a great view of the lower fall. It's a LOT of steps, and descends about 500 feet - and then of course you get to climb back UP that 500'.
Between Canyon and the Yellowstone Lake area are the Haydn Valley (another wildlife spotting area) and the Mud Volcano area, where there is a short trail with big mud pots and some pretty hot spring pools. In the Lake area we enjoyed hiking the Storm Point trail, Elephant Back Mountain, and the trail to the Natural Bridge.
I recommend the West Thumb Geyser Basin - there aren't really any geysers there, but it had some of the most beautiful pools that we saw anywhere in the park.
There's tons to do in the Old Faithful area. Make sure you do the Upper Geyser Basin all the way out to Morning Glory Pool. If you have a chance to see Castle Geyser erupt, then do so (it's generally twice a day, and it goes off for 45-60 minutes!) - it's much more impressive than Old Faithful. Daisy Geyser is also very good. We also walked around Biscuit Basin and Black Sand Basin and hiked to Lone Star Geyser.
If you drive west from Old Faithful there's the Midway Geyser Basin,the Lower Geyser Basin, the Grand Prismatic Spring (VERY colorful!), and the Fountain Paint Pots. We hiked to Fairy Falls. If you scramble up the hill that's near the beginning of the trail to Fairy Falls (maybe 1/4 mile in) you get a great overhead view of Grand Prismatic Spring, but be aware that it's very steep and there's no real trail (though lots of use trails from other people have done it), so you have to be careful.
I haven't been to the Tetons in years, so I don't really remember what there is to do there, sorry.
Hope that helps!