Mostly I say the name of the store, but if I am speaking to someone who is not familiar with the store I say supermarket, as in, we went to a supermarket during vacation, instead of saying we went to Harris Teeter, or Piggly Wiggly- which do not exist in New York City. When asked about my career, I tend to say I work in the supermarket industry, rather than naming the chain.
I never say grocery store, as "grocery" is just one part of the store to me- the grocery section, or grocery dept, as opposed to the dairy, frozen, meat and produce depts.
I also say cart or wagon, I've never heard of it referred to as a buggy until I visited a supermarket in another state. Even wagon is not a widely used term for a shopping cart, as I got some odd looks the first time I said it to an employee when I worked in a store that was further south in NJ than I had ever worked.