Hi, U.S. native speaker here. We use SHOP as a noun, but only as a synonym for STORE. Also, it's not a rule, but we are more likely to use SHOP for smaller independent businesses and STORE for larger ones. For example, there is a small GIFT SHOP, or CHOCOLATE SHOP in my neighborhood. But no one would call a large store like wal-mart or IKEA a shop. The equivalent of WEEKLY SHOP as you use it above would just be WEEKLY SHOPPING. Also, in answer to another answer - - a convenience store is a small store - the sort of store that's part of a gas station and is open all the time. A supermarket is a large grocery store.

In Canada we would say "I do my weekly shopping on Mondays." It's more common to use "store" instead "shop" as a noun, as in "I go to the store." (Another casual saying here is "I do my weekly grocery run on Mondays.")

I have looked it up, and 'grocery store' would be a better alternative. It gives you some sense of the difference. However, like I already mentioned, a native American-English speaker should be able to clarify.

I am in no position to give a confident answer since I am a native Dutch speaker, but as a person who watches a lot of American tv-series and movies, I would have to say 'store'. The expression 'to go shopping' seems more British to me, while an American would say 'to go the store'. If I'm correct, Americans tend to say 'convenience store' instead of 'supermarket', as is more prevalent in (native and non-native) English spoken by Europeans. I hope this helps.