If it were my trip, I'd pick a few different places to visit, but this is based on my tastes and often to get away from the crowds. I'm also assuming it would not be a winter trip, where some of these roads would be dangerous to take. I would visit Bear Lake, on the northeast side of Utah, and shared with Idaho. The color of the water is stunning, although it may be a bit cold for swimming in. Stop in Garden City for some raspberry shakes, then head east down Logan Canyon. Take a side trip up to Tony Grove to see this beautiful little lake and take a quick hike around it. There is a nearby Forest Service campground if you want to stay overnight.
After reaching Logan, continue to head east, towards the Golden Spike monument (and the Spiral Jetty). Head north and west , then south around the west side of the lake, where you are on your own, out of touch with the rest of the world for mile on end. Pick the right dirt roads, and you can end up on the Bonneville Salt Flats, where timing could help you catch cars going for top speeds, or amateur rockets going miles and miles up into the air.
If you cross over 1-80, you could visit Blue lake, a 50 foot deep, geothermally heated lack directly south of Wendover. Not the prettiest views around the lake, but a unique site in the middle of the desert. You can then head south east towards Fish Springs, another set of lakes in the middle of nowhere. In addition to the fish, there are lots of birds that use this as a migration stop. This will also put you on the Pony Express Trail, where you can head back east towards Simpson Springs. Again, you are essentially on your own, but there is better cell service, as Dugway Proving grounds (military base) is covered. If you are anxious for the trees you haven't seen for 200 or so miles, you can find some in the mountains just south of Simpson Springs. If you are bringing ATVs, there is a fantasic ride through these mountains that will leave you doubting the concept that this is a desert.
At several locations along the Pony Express trail, markers and buildings commemerate the Pony Express. If you venture a little further south on the dirt roads,almost to Delta, you can find the Topaz Internment camp, one of the locations Japanese families were sent during World War 2. There is a small museum where the history can be explored.
Leaving Delta, you can head northeast to Provo, Utah and up Provo canyon. Here you will find Bridal Veil Falls, with a nearby campground connecting via a paved walking/cycling trail.
If you travel up Provo Canyon a little further, you can take a left on the Alpine Scenic Highway. Travel this road for a while, and you will reach Cascade Springs, and amazing site where the water comes up out of the ground, and cascades it way down the mountainside. Beautiful site, with walkways out through the water and bridges over some of the waterfalls. Plenty of campgrounds in the area, although very few have hookups. While you can continue on the Alpine Scenic highway, I wouldn't recommend it with a camper, as the road is narrow and twisty. There are length restrictions on this road for good reasons. Instead, you can head east from Cascade springs towards Midway, Utah, then towards Park City, where it is worth stopping to look at the Olympic venues. You might catch ski jumpers going into the pool.
You can then head down into Salt Lake City, where a visit to Temple Square is encouraged. Heading north, you will find Antelope Island (but be aware that there are tons of bugs in the summer). You could then head west along I-80, back through the desert towards Nevada. If you can stay awake, you will even get a chance to glimpse the one "tree" along the way.
Back in Wendover, you can visit the airport where they practiced for the flight to drop the atomic bombs on Japan. On the Nevada side, you can catch a concert or do some gambling. At that point, I'll leave it to someone familiar with Nevada to guide your route.