Having just come back from Yellowstone there is no point in going for just two days to both YS and Grand Teton. It will take you the best part of those two days just driving to and from the park and then through it. You're basically not going to have any time to actually see anything, be that the sights or wildlife (other than Bison).
Getting from Mammoth to Jackson will take you the best part of 4-5 hours (non stop) just getting from the north of YS to Jackson, and at least an hour (if you time it right) to see Old Faithful. Most of the geyser basins take an hour or so to walk round as well, let alone if you don't just want to take the most direct route (i.e. see Hayden or Lamar Valley). We did the main sights of Grand Teton in a day, but I'd recommend at least two whole days in Yellowstone on top of that (+ one more for travel to yellowstone and maybe another from Jackson to SLC and Provo) - so 4 days minimum really. We spent a week there and would have liked to have spent more time in Grand Teton.
If you do intend to still do Yellowstone, book early and if camping book a pre bookable campsite, otherwise prepare to be waiting at the first come first serve campsites at 6:30 am to be in with a chance of camping there that night!
Also didn't realise how liberal the US was regarding speed. The main highways had limits of 80mph and most of the more major single lane roads were 70. I don't remember them being that high last time I was there (about 5 years ago).
EDIT: To join it up with posts a couple of pages ago, we did it in an F150, along with about half the people there (well pickup truck of some sort). Also stayed at Glacier last night on the way back. Road to the sun is still closed (obviously) but the scenery is stunning there and the wild flowers are out which really adds to it!