I usually say supermarket, but if I'm talking about the small supermarket around the corner I sometimes run to I just say "I'm running to Costa's," even though it's a supermarket too.
Grocery store here. We go grocery shopping. I also think it's a regional thing. I'm originally from the Chicago suburbs and now live in Green Bay, WI. It's always been grocery shopping in both locations.
I say grocery store, but mostly now, as we tend to shop every other Friday night, we just say we're going shopping or going to Target and Kroger. Both of my DD's say we're "going to shopping" LOL
I say buggy, too! A couple of friends who relocated to my office a few years ago are from Missouri and the first time they heard me say buggy they just died laughing. She said "It's a CART! And I was like, no, BUGGY!" I still laugh thinking about that conversation.