N.W.
I used to live in Park City and liked camping in the national parks: Zion, Bryce, Arches, Snow Canyon near you in S.Utah. It can be hot in summer and Bryce would be the coolest then. Jordanelle reservoir near Park City has a lovely campground and is cool in summer. Make reservations.
If you stay at one of the developed campgrounds, they should have bathrooms with hot showers, and plenty of running water. You can also check to see if they have bbq pits or grills so you can bring skewers to roast hot dogs and make smores.
Some camping tips:
I plan cereal and milk for breakfast and sandwiches for lunch. That way I only have to cook one hot meal a day.
Prep meals at home. It's harder to chop and mix in the great outdoors. Precooked foods like hot dogs are faster and easier to cook than raw foods like hamburgers and chicken.
Paper plates. It's a pain to wash dishes when you're camping.
Freeze a couple of gallon jugs of water to use as ice in the cooler. Pour a little out first because the jugs will expand in the freezer. Once in the cooler, they don't melt as fast as a bag of ice, and you can drink the ice water as it melts.
Some stuff I like to bring: Flashlights and a rechargeable lantern, plastic tablecloth, beach chairs and a beach umbrella, buckets and shovels to play in the dirt but look out for scorpions in the desert, they blend in with the sand. I used to have one of those open canopies that was like a tent with no sides. It gives you a large shady area to sit in during the day.
Don't leave your shoes outside the tent at night. If you do, shake them out first before putting them on to make sure there are no critters in them.
We snack alot when we camp. All that fresh air makes us hungry, so I bring more cookies, crackers and chips than we normally eat.
Freeze a bunch of juice boxes or packs before leaving. It'll help keep the cooler cold and after a day or two, they are a nice slushy consistency.
Keep the cooler in the shade and cover it with an extra blanket for better insulation.
You might want to try a one night practice run with the camping equipment before you go on a long trip. That way you'll see what you don't need to bring as well as find out what else you need to buy to be comfortable for an extended time.