On my side of town it isn't just English, so I shop in grocery stores and supermercados. The huge stores like Target and Walmart are more than groceries, so they tend to call themselves "super centers." Costco and Sam's Club are warehouse stores. So it looks like the closest to "market" around here is the mercado. Most stores are known by name, so I would never say "I'm going to the supermarket" - it's "I'm going to Kroger" (or Albertson or Aldi or Fiesta).