Be sure to have proper tie-downs on your truck. For years I used the bed stake holes, but on long trips they'd often wobble just enough to loosen the chains a bit. Once, while driving in strong and gusty wind conditions on a thousand-mile trip, both of the driver's side tie-down chains got so loose that they detached. I came over a hill on I-35 in Iowa and glanced in my driver's side mirror to see the camper lifted up about two feet and still going! I slammed on the brakes and made a fairly hard right turn onto the shoulder and it settled back down. (Whew!) I drove the rest of my trip at about 30 mph and purchased a set of good, frame-mounted tie downs.
I could go on and on
...like the time I drove the pickup to my insurance agent's office to turn in a claim (from backing my truck camper into my car) and smacked the overhang/sign on the front of his office. (One of my more embarrassing moments!)
Or the time the refrigerator door came open enroute and a jar of pickles fell out and crashed onto the floor. (Pickle juice is STICKY!)
But I won't. ;)
I saw one of those a few years ago owned by the owner of body shop. I like it and have suggested here that Ford, GM and Dodge could offer such beds on new trucks. Trucks seem to get taller every year, and 5th wheel manufacturers can't seem to keep up with them.
I can't for the life of me understand why everyone is recommending I-80, ESPECIALLY when you've stated that you'd like to visit the Black Hills.
I would recommend I-90, probably after crossing Iowa on I-80/680, then taking I-29 north to Sioux Falls, SD/I-90. Traffic, especially truck traffic, will be much lighter on I-90, and it will take you directly to the Black Hills (still an excellent vacation spot). I'd also recommend a drive-through tour of Badlands National Park, 60 miles east of Rapid City. Driving through (possibly with a short hike) won't slow you much -- maybe figure an extra 2-3 hours including a stop at Wall Drug, the granddaddy of all tourist traps. ;)
The Black Hills has plenty to entertain for a week, but I'd think you'd want to pare that down to 2-3 days to leave more time for Yellowstone/Grand Teton NPs. Rather than recommend what you should see there, I'd suggest that you Google "Black Hills" and decide as a family what you want to see. Certainly Mount Rushmore, Crazy Horse Monument and Custer State Park should not be missed, but whether you want to include one or both of the NP caves (Jewel and Wind), Deadwood, Devils Tower, etc -- that depends on time available and interest.
Continue on I-90 west to Buffalo, WY, then take U.S. Hwy 16 across the Bighorn Mountains. It's a good, all season highway with not too many steep hills nor sharp curves. Continue on to Cody where again, you can choose from rodeos, the Buffalo Bill Historical Center, etc.
Sorry, I've got to go, but that'll get you started -- and nearly to Yellowstone.
I'd also recommend making the reservations at Madison Campground. It's about as well centered as any. In fact, if you need hookups I'd point to West Yellowstone (or further), which is another ~20 miles west of Madison.
As others have said, start out early in the morning. I used to visit several times each year, and we'd be out of camp before sunrise, then back in camp at 10 or 11 a.m. until 3-4 p.m., then back out until dark. We were concentrating on photography, and the best photos usually are found in early morning and late afternoon. Just so happens that's also the best time to avoid the mid-day crowds/traffic. :)
If you won't be driving the truck much, high mileage is probably okay, but expensive stuff does go kaput on these high mileage trucks, and I would STRONGLY recommend you not plan to drive a high-mileage pickup too much.
I traded off my 2000 F250 PSD 4x4 a year ago following the second year of $10K repairs! I bought it new and only drove it 6K-7K miles per year for the first decade, but then I took a new job and drove it 30K miles per year for three years. It was nothing too major (clutch was the most expensive), but it was a host of $1,500 repair bills. Seemed I couldn't go for two months without some $1500 repair job.
It had 151K miles on it when I traded it, and it needed another $8,000 or so in repairs -- turbo waste gate, clutch again (damaged this time by a leaking waste gate), 2 injectors and some routine stuff that was about due, like front suspension work and new brakes. It was clean and rust-free. I got $3,500 for it in trade. I figured a brand new one would have been cheaper. (If you're a DIYer, repairs wouldn't have been so bad, but I always took it back to the dealer.)
My extensive repairs began at around 100K miles. Had I continued to drive it 6-7K miles per year, the repairs would have been manageable at $2,000 or so per year, but multiply that annual mileage (and repair bills) by five and it wasn't feasible to keep it.
I also wanted to recommend that you get a good tent with an adequate rain fly. I bought a new tent several years ago and barely got it up before a nice summer storm blew in. Thankfully I hadn't gotten my sleeping bag or any clothes into it, because there was half an inch of water standing in the bottom of the tent by the time the storm was finished -- all having come in through the windows since the fly didn't completely cover them.
I'm probably guilty of some overkill with the tent we now use -- a Cabella's Alaskan Guide tent (iirc). It's a great tent (6-man), but it takes both of us a half hour to set it up. That's after some practice! But don't under do it either. A tent full of water, soggy clothing, sleeping bag, etc. can take the fun out of camping in a hurry.
Your coldest nights will be in Idaho, Montana, Wyoming and Colorado, especially if you camp in the mountains, where most of the NFS campgrounds are found. We meet family in the Bighorn Mountains of Wyoming each August. We camp at about the lowest campground on the mountain so it'll be warmer, but the night temps always drop into the 30s or 40s. This is at about 8500 feet ASL.
In case you weren't aware of it, the standard atmospheric lapse rate for temperature is 3.5F per 1000 feet, so if you had an 8500-foot mountain in Florida, temps at its peak would be ~30 degrees F cooler than those at sea level. Since the plains in the Rocky Mountain West are already at around 4500 feet elevation (+/-), temps at 8500 feet will only be ~14 degrees F cooler than at the base. However, the thin, cool, dry air of the Rocky Mountain states also means there will be more of a temperature drop in the evenings, after sunset, and this is also exaggerated with higher elevations in the mountains.
Where I live on the plains, evening summer temps normally drop 25-30 degrees, so on a typical day with a high of 85F, night temps will drop to 55-60F, and at "our" campground in the Bighorns, it'll drop to around 40F.
As you might imagine, ground temps in the higher elevations are pretty low, and that temperature will be quickly transferred to your air mattress, which will in turn be transferred to YOU.
All this is simply an explanation of why you'll need some kind of insulation between you and the ground, and an air mattress provides almost NO insulation.
For our August camping trips, my wife and I use a double thick (22") queen-sized air mattress topped with a couple -30F sleeping bags zipped together at the bottom. One (spread out) bag stays under us for insulation from the mattress, the other is thrown over us to use as desired. Usually I don't cover up completely until the wee hours of the morning when the outside temps are at their coolest. Our dog, a silky, sleeps on top of the sleeping bags until about that time too, when she starts shivering and climbs under the top bag to cuddle with us.
We like the thick air mattress because it sits high enough that we can sit on the edge of it, whether climbing into bed or getting out of it. At our ages (60 and 70), we like all things that we sit on to be at butt level!
While on the subject of air mattresses, I've found that at higher elevations the little $30 DC pump doesn't work too well. It'll get the mattress almost full, but a better pump is needed to top it off and give it the firmness for a comfortable sleep. I have a converter in my car that works well with the mattress' built-in AC pump ; otherwise a hand or foot pump should work. I plan to get one next spring.
I live near I-90 in eastern Wyoming. Our standard trip to Yellowstone is over the Bighorns (Hwy 16 or 14), then through Cody and the East Gate. While I prefer to drive through Yellowstone bobtail, there are lots and lots of others driving through with their campers, and I can do it too -- and enjoy it.
That's what I'd recommend to you. Hwy 16 leaves I-90 at Buffalo and is a fairly gentle mountain drive across the Bighorns. Hwy 14 leaves I-90 about 50 miles NW of there at Ranchester and is a little steeper going up and coming back down from the Bighorns, but it's not hairy-scary. Both are about the same distance, whether measured by miles or time, and both are good, wide, scenic, all-season highways. If you choose Hwy 14, DO NOT take 14A at the top. It's too steep for your rig, imho.
This recommendation comes with the assumption that you're fairly experienced at handling your rig (gathered from your profile) and that you've done some mountain driving with your rig (w-a guess). :)
Here's another in favor of using 4LO, especially coming down! Inching down a steep trail with 20K+ pounds can be too much for your brakes. (Ask me how I know.) ;)
Yes, I did it once. I had a 2500 lb. slide-in camper on an F250. I'd been down that 4x4 canyon trail dozens of times before, but this time my girlfriend (now wife) was riding along. She's scared to death of steep and narrow mountain trails, so when she started freaking out I slowed down. I don't remember whether I was in 4LO or 4HI, but I'm thinking it must have been 4HI, as I rode the brakes a little too much... and suddenly they were GONE! Luckily I was almost at the bottom of the canyon when they left me, and I was able to find a place where I could leave the trail and head up a little hill until we came to a complete stop. Were it not for that little hill, I'd have been plowing into cars, people, or maybe a very swift stream.
And you're talking about doing it with a 14K trailer. Don't do it in 4HI! If this is a 25% grade, you need 4LO. This is why you have it.
As for the front tires grabbing in a turn, let 'em. On a dirt road it's not going to hurt a thing.