I have been twice there, took my first archery antelope last year. You will want to be in either the NW corner or the SW corner of the state, either can be ok. Antleope took a pounding and everyone said don't go, they even cancelled the non-res rifle hunts last year. I had plenty of opportunities, its still just about the hardest animal in North America to get an arrow in spot and stalk.
Environment wise, you are never going to get close on the flat open plains. You need to look for broken terrain that provides the opportunity to put a hill between you and the antelope while you get close. Patience pays too, if you are in a good spot and they are coming your direction, wait them out. You can find this kind of terrain as the high plains break up going down to a river. This same envirnoment can hold good stalkable mule deer as well.
You will get shot opportunities in the 55-65 yard range about 3x as frequent as inside 40 yards in my experience. If you have your bow dialed in and can shoot those distances, your odds of success will increase.
There are more hunters in the SW corner than the NW corner. You will have the most opportunity in large tracts of walk in area. You can camp on BLM lands I think, but check with the game and fish first.
Spot and stalk antelope is a lot of walking all day. No way around that. Be ready to put 17-20 miles a day on your boots and you should be able to have opportunities from sun up to sun down.
Where to start - look at this antelope density map:
http://gfp.sd.gov/hunting/big-game/images/antelope-thunderstormmap.jpg
Next find public ground in the areas with the most goats. SD has a pretty good online atlas, I keep a printed copy with me when i'm up there.
http://gfp.sd.gov/hunting/areas/
Contact the offices for the areas that you are interested and have them mail you maps, its like $8-$10 per map usually.
Do some online research via msg boards, etc. Everywhere you find online to go will be way overhunted, but it can tell you what you want to avoid.
Everyone in SD was very freindly and supportive of hunting at the restaurants and bars. It can be good to stay in town, you get a lot of tips and information from locals that you can put to use if you don't have an area figured out.