Lots of people are doing it. Some in RVs. Some drive across country in their own cars. Others, like us, pick a starting point and rent a car. Whatever you chose, be sure to chose your time of year carefully. Too early or too late and you will run into cold, icy, and even snowy weather. When we were leaving Bryce Canyon on September 27, they were expecting snow for the next morning. The elevation is higher and the weather turns sooner. Of course if you like snow. . .
Middle of the summer is going to be a hot time in the desert areas like Arches. Of course as any traveler knows, the weather is unpredictable no matter where you go. We were lucky to squeeze our trip in mostly between weather fronts created by two hurricanes off the coast of Mexico that brought the flooding rains one day of our trip.
If you want to stay at a lodge in the parks, especially Zion, book early, early, early. Bob was planning our trip about six months in advance and couldn't get into the Zion lodge.
There are little "hotel towns" that have sprung up outside the entrances to the parks. Springdale outside of Zion and Bryce Canyon City just outside the park entrance (you can catch a shuttle into the park from there). Moab is just a little farther up the road from the Arches entrance and at the southern entrance to Grand Canyon just south of the Village is a town called Tusayan where there are several hotels about 20 minutes from the Village. All the towns have some national chain hotels and restaurants/fast food places if you want to save some money on lodging.
The dress at the parks is casual. Everyone is in jeans and shirts/jackets with backpacks with the exception of a few who probably arrive on buses and need to make a fashion statement wherever thay go. Those are the ones who won't be hiking the beautiful trails.
Stock up on bottled water or take a refillable bottle. Most of the parks have spring water at the visitor center or the restroom areas where you can fill up. Be sure it's marked spring water. Don't fill up from the faucet in the restroom.
Dress in layers. Always a good rule no matter where you are traveling. The mornings and evenings will be chilly. The clear skies while beautiful in the daytime, don't hold the heat when the sun's not shining.
Did I mention hydrate often? Utah, Arizona, and New Mexico are arid climates. You will need to drink lots of water. Use a little Vaseline to swab your nose if it gets too dry and if you tend to have dry skin anyway, be sure to take some moisturizer.
Sun screen and bug spray are good to have. We only needed the bug spray once but other times of the year I'm sure would it would be even more necessary.
Take a soft sided cooler to carry sandwiches, milk for cereal, juice, etc. Almost every place we stayed had a small refrigerator. And if they didn't, there was always an ice machine somewhere. I take a couple of gallon sized Ziploc bags for the ice.
Be safe! Stay on the trails. There are a lot of tempting overlooks that have no safety rails and it's hard to know what's supporting the rock from below--if anything. We held our breath often as the crazys ventured out to small precipices to pose for pictures. One even balanced on one foot in some kind of yoga pose. Amazingly, only about 12 people fall in the Grand Canyon a year and they only register a couple of deaths. Don't be one of them.
If you're not a big eater, you might consider splitting a meal in the lodge restaurants--especially Bryce. The plates were huge there. You can always order more or it will give you an excuse for desert as well. Sometimes there was a charge for that and other times, not. Ask.
That's all I can think of now. I'm putting Bob to work planning the next one. We'll do a more northern route, even possibly driving from Ohio and back. On the list: Yosemite, Yellowstone, Grand Teton, and of course a visit to the stone presidents.