Hill City, SD to Fishing Bridge in Yellowstone in mid-June

X5
I have driven 14, 14A and 16. 16 is very doable in an RV. 14 is not too bad since the middle part East of Burgess Juncrion was rebuilt, but is still steeper than 16. 14A is a definite no in an RV once you get West of Bald mountain where it rapidly drops in elevation.

X4. Grades on 16 are not that steep and not as many curves as 14.

Agree with 2 gypsies. We traveled the same route last july in gas truck towing a tt with lttle problem. We are also from south florida with little mountain experince so i had the same reservations as you. I think that by tbe time you get to that area you will some pratice and will be more comfortable. Monteagle and the Vail pass bothered me more than the the Buffalo, Grey bull route did. Just remember the basics of mountain driving and relax.

X2 on 2gypsies post. We have used route 16 once and route 14 (not 14A) twice heading eastbound and route 16 does have shallower grades. As long as you take your time and use lower gears when going down you will be OK. You will have a good climb after going through the east entrance to Yellowstone, so the climb up route 16 will be good practice.

I see no issues with taking the route you have planned. It's very doable in a RV. You'll have Powder River Pass to cross so you'll have a climb from Buffalo and a descent to Worland. By the time you get from Florida to Wyoming you'll be an old hand at driving these roads. You'll also have some of it in Yellowstone itself so this will be good practice. :) Just know how to brake your RV correctly. Downshift before you even start going down not when you're midway. It's short and it'll soon be over! We've done it often with our 40' motorhome and many RVers do it.

We would not do Hwys 14/14A.

I think the others are guessing you want to drive Hwy 14 and 14A. Your plan doesn't include that stretch.

This is the route you want:
From Buffalo take 16 to Worland and north to Greybull and west to Cody.

If you want to explore on Hwy 14 or 14A, stay in Greybull and take your vehicle.

https://www.wyomingtourism.org/things-to-do/detail/Cloud-Peak-Skyway-Scenic-Byway/7344

To get to this point you climb 4,000 ft in about 15 miles. The signs say 8% next 10 miles on the way down.

As said previously, 14 takes you up, and down, very steep mountain pass. The same with 16.

90 will take you up Ranchester hill just north of sheridan. Kinda steep, but tame compared to 14 and 16. From there it's pretty smooth sailing with some small foothills until you turn south at Livingston on 89. There you follow the Yellowstone through Paradise Valley. It's a cake walk compared to 14/16, but a lot longer.

Both routes are steep mountain passes that top out at over 9000'. If you are leery of towing through mountains this would not be my choice for a first try especially with a gasser. And remember going down is worse than going up. I-90 to Livingston then south on 89 to Gardiner is mostly flat and a beautiful drive that follows the Yellowstone River most of the way after Billings.