Having grown up in the Black Hills and still living here, I can tell you that almost any campground will be good in one way or another. It will just depend on what you have and what you want.
I prefer the state campgrounds because they tend to be the best for getting in and out easy and the fees go to support keeping our state parks looking good. My favorite is Deerfield Lake just outside Hill City. I absoultely love the evenings up at that campground. The fishing is pretty decent as well. Mostly Brown and Rainbow Trout in the hills. Many of the campgrounds already mentioned are very good as well. I can't think of any that do not allow pets. Some do have a little bit more restrictive rules on where you can go with them though.
For those who are wanting to avoid the Rally traffic, the Sturgis Bike Rally is always the first full week of August. So plan accordingly.
Mt. Rushmore and Crazy Horse have of course been mentioned. Custer State Park is awesome, check out the Needles near Mt. Rushmore. Wind Cave National Park and Badlands National Park are in the area and provide dramatic scenery. You will have a hard time believing that there is such vastly different terrain so close to each other. If you like to fish there are a ton of places to try. My favorites are Horsethief lake near Mt. Rushmore, Deerfield, Pactola, Angostaura and Orman. For stream fishing Rapid Creek is good depending on where you can get into it.
Devil's Tower is in Wyoming but has some beautiful scenery. If you are going to go over there, drive up to Belle Fourche and take the back highway in. It is actually a little faster than taking the interstate to Sundance and much more beautiful.
If you want to see a really beautiful city camground with quick easy acces to stores and food, try the Spearfish City Campground. I live and work in Spearfish and it will take your breath away to come into the city and then find this beautiful quiet campground nestled so close to downtown.
For any of the TDR members coming through, drop me a message on here. I am always up for trying to get together with a fellow diesel enthusiast.
Should you have any diesel related needs, Rapid Diesel in Rapid City can take good care of you and Scott Peterson Motors in Belle Fourche has always been really good to work with. Belle Fourche also has Lindstad Alignment, they make an adjustable track bar for the 2nd Gen Dodges that is top rate. I have had one on my truck for years with excellent results.
Overall, if you come, I think you will really enjoy it. The Black Hills is an awesome place to be year round and I can't imaging ever wanting to be anywhere else.
Jon Tish, aka Cowboy Medic