Kevina,
Some great advice from the others. This will be my 4th trip to SD this fall and except for one day that I paid to hunt some private land, it's all been done on public land.
Yes, you will see other hunters but there is enough land available for you to have a great hunt. My hunting party usually consists of 2-3 guys. Smaller, out of the way areas work well for us. We probably hunt waterfowl areas (near food) as much as roosting cover and food plots. Now we don't see hundreds of birds every day, but for the most part, if we don't limit out it's not because we didn't have a chance to...just lousy shooting on our part.
Like the others said, get the SD atlas and check out the public land available in the area that you are staying. With a little scouting, you should have great time in SD. I'm taking my 1 year old lab out there this year so I'm like a kid waiting for Christmas.
Have a great hunt and let us know how you do.
Scott