I've made many trips to SD over the years, hunting exclusively public land. I'll just add a few things to what's already been said.
The publication you need is the SD Hunting Atlas:
http://www.sdgfp.info/Publications/Atlas/Index.htmif you know what part of the state you'll be hunting, you can just print the pages with a color printer, otherwise they'll send you the atlas for free. I see that since I last hunted SD in '06, they've added the school lands to the atlas (in blue) which is a nice improvement - this can be hunted as long as the crops are down, but in my experience a most of it is grazed and it's rarely worth a long drive to check out a piece of this ground.
Do not overlook the small parcels, and hunt the isolated or harder to reach parts of the larger parcels - most hunters tend to follow the same general routes and there's enough pressure that the birds get trained very quickly.
Don't beat your dogs up hunting large pieces of CRP grass during the middle of the day, unless the crops in the vicinity have recently been cut. If you see lots of pheasant sign in an area of grassy cover, go back there the last hour of sunlight. This last hour of the day has been the most productive for me over the years as the birds will move into the roosting cover they like even if it gets pounded all day. Unfortunately, they seem to have access to the sunset tables, and on a number of occasions I've seen whole flocks head out of the crop fields and into the walk-in or public ground a few minutes after sunset.
As the season progresses, the birds will spend a lot of time in cattails, which is a type of cover you'll have plenty of access to as a public land hunter. Focus on the smaller sloughs - the big marshes will wear you and your dogs out fast, especially if it's a wet year and it's before freeze up. In this type of cover, give the bird the second barrel if there's any doubt at all, and mark the fall closely - the dog often can't see the fall, plus it's so noisy that they may not even when the bird flushes. If there are 2 guys the one who shot the bird should stay put and direct his parter and the dogs to the fall. Even with good dogs you will lose some birds. After things freeze by all means try the bigger cattail sloughs, especially if there's some fresh snow.
Do hunt the public rights of way. I think this is probably the most productive way to kill birds if you don't have private land access, especially in the difficult period in between the first week or so when there are dumb birds everywhere, and the latter part of the season when weather concentrates the birds in the heavy cover. The best roads are the nearly abandoned section roads that have nice cover on both sides from the fencelines right up to the wheel ruts. These are usually marked with dashed lines in the atlas. If you have two vehicles leave one at one end and drive around to the other end to start. If not, hunt the mile, have a snack and take a rest, and you may well pick up another bird on the way back. In particular, hunt ditches that border cut crops. As noted above, don't shoot near buildings, and don't stray outside the right of way even if there's a marshy area with no fence (it can be tempting). Try to keep your dog out of the fields as well. This is a good way to hunt in the middle of the day when the birds are loafing in light cover near the food source.
Unlike the guy hunting private land, you won't have the luxury of sleeping in. Get up early, burn some gas scouting the public spots, and park at your first field well before shooting hours (in heavily hunted areas, particularly weekends, this means a couple of hours before shooting starts to avoid finding a couple of parties ready to go at every public spot you drive past).
Don't expect limits every day, and the probability of limiting out is inversely proportional to the size of your group in this kind of hunting. You will work hard for your birds, and you'll remember every one, which may not be true for the guy paying big money to hunt with a crowd, shooting birds out of strips of milo with walking paths in between. I think you'll enjoy it!
If you haven't decided what part of the state to hunt, send me a PM and I'll give you some suggestions.
