Yellowstone camping at it's best - Madison
First of all do not expect any internet connection.
After that enjoy good sites, clean bathrooms, and excellent fire pits.
This campground is 14 miles from West Yellowstone-- or "West". Generally in the park you will double the mileage given for time of travel. So, 14 miles away from West means it will take you 30 minutes unless there is a bison jam, etc.
Once you arrive (and if you think of it while making a reservation) you can request a loop. Tent campers prefer the G & H loops while RV people gather in the others. It's worth trying to state a preference. The H loop is nearest the Madison River. Both G & H are largely shaded although this is not true for 292, 272, 273, 274.
276-286 are best in G loop. There really are no bad sites in Loops G & H other than the four above.
For the past decade I have camped in Madison each June.
Wildlife tip: If you want to see wildlife in YNP visit the park in late May to mid June--- otherwise the mammals go up into the high country for the hot days of summer and effectively vanish for most visitors. In the Lamar Valley they'll stay through June.