I've hunted them in WY but never in SD.
It was pretty easy there at that time back in the 80s. The herds tended to be creatures of habit. The locals pretty much despised the "goats" and were happy to give us the inside info. A herd used a certain school section to graze about the same time every day. We were instructed on which point to set up and the goats would come by about 0900. By 0915 the tags were filled.
The were a few places like that and, as I said, the locals were only too happy to help you out and grant permission.
We threw a big game feed on the last day of our trip. We provided all but the desserts and asked the womenfolk to bring those (very smart move; they all tried to outdo each other). We had antelope backstrap and deer backstrap wrapped in bacon, fresh trout from the ranch streams and did some chicken. Taters, beans, salad stuff. Funny part was all those ranchers that "would never eat goat" ended up having seconds on the antelope backstrap. Now, admittedly, these antelope dined almost exclusively in alfalfa fields so the overpowering sage taste wasn't in it. Anyway, it ended up being an annual thing for about a decade. Usually 30 people, trap shooting, eating, drinking. Good times.
My advice? Figure out where you want to go and make some contact with the locals. Antelope can demolish a big hay stackyard when it's cold and snowy. Might be some ranchers willing to let you thin the herd.
As far as equipment, I used a .270; worked great with 130 grain PSP. Decent camo clothing is a good idea. A good set of binocs. Maybe some sort of lightweight camo cloth on stakes to help make a hide. I never used an antelope decoy; never needed one. Good boots, always.