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.