OK we live in North Central Florida and over the years have made that trip a number of times. We just got back from taking the first section up to Iowa.
I-75 -> I-24 in Chattanooga
I-24 to I-57N
I-57 to I-64W
Then to avoid St Louis which in our opinion has the worst roads and traffic of any major city.
We take Ill Hwy 127 north to I-55 then I-55N to I-72.
Get on I-72W into Iowa then US 61N to I-80.
Follow I-80 to Cheyenne WY where you get on I-25N.
At that point you can get into Yellowstone at least three ways. US20 at Casper WY. US 14 at Sheridan WY or follow I-25 to I-90W and then south at Bozeman on US 89. Somehow you have to climb to 8000 feet at Yellowstone. We haven't taken the US 20 route but have done the other two and I don't recall any extreme grades on either one. We did the US89 route last summer and it seemed to be fairly flat until you get to YNP. Bozeman is at 5000 ft so there is a 3000 ft gain in there.
Do not drive the Beartooth Pass since it does have steep grades and lots of switchbacks.
As for how long you drive each day, that depends on your schedule. If you are short on time them you need to drive longer each day obviously. We used to do 500 mile days before I retired, now we aim for 300-350 mi/day. Doing 500 mi days should see you in YNP in 4 days. Depending on where you begin.
I saw someone mention Pensacola, if you start from there I'd take I-10W to I-55N then follow that to I-57N. From there just follow the above route.
I fudged on the section near St Louis we actually get down on the smaller roads which are fine but twisty with lots of small towns and drive along the Mississippi to Keokuk IA then take surface roads thru Iowa. But I'm not sensitive to grades and curves.
