Translation:I go out to the convenience store every morning.
Duo's translation is worded that way because they're trying to convey in English the meaning of 出かける - to go out. Duo does this a lot - tries to make literal translations of the Japanese work in English. Personally, I like to avoid "translationese" as much as possible : )
Yes, a convenience store is a specific type of store. But I almost never use the full phrase "convenience store" when I'm talking about a regular store that I go to for buying food or groceries (since it's going to be either a convenience store (also called a "corner store" sometimes), grocery store, or supermarket). "I go to the store every morning" is what I'd say. Another example: "I'm going to the store. Do you want anything?" I've only ever used the full phrase "convenience store" if I'm talking about different types of stores. If clarification is needed, I usually use the name of the store.