To me, the 'Date Range Availability' tab is the most useful on both Recreation.gov and ReserveAmerica.com. To repeat myself from an earlier post, when you make your reservation on one system, your dates are reserved immediately and displayed on the other system within 24 hours.
After you have found the campground you want, enter your expected arrival date. Next, enter the number 1 in "Length of stay". You can change the number of nights to reserve later but this helps you see the most sites available on the availability calendar. Click the "Search" button to display the Date Range Availability schedule. I'll explain how the sites are sorted, and how to change it, below the example. On ReserveAmerica, you'll need to scroll down below the map to see it but it looks similar to the Recreation.gov example below:
Campsites are not listed in order of site number initially. They are listed in order of availability on the arrival date; based on letters described in the Legend: A, W, X, R.
You can change the order displayed by clicking on the column heading "Site#" as shown below:
Click Site # again and it will sort in reverse order.
Find other tips for using the reservation system here: How to Make a Reservation