The verb form is always "to shop," "shopping." We go to the store to do our shopping.
We use both "store" and "shop," it is just a question of frequency. "Shop" carries an idea of a small place, a boutique. We have an expression or idiom, "a mom and pop shop," which means a small store owned and operated by a couple, or perhaps a family. We might say "the hotel has a gift shop in the lobby."
Stores will sometimes use the word "shop" in their name in order to give an impression of folksiness or friendliness.
Looking at an online business directory, I notice that they list these as categories (among many others): Department stores; furniture stores; gift, novelty, and souvenir shops; grocery stores; hardware stores; hobby, toy, and game shops.
An interesting entry is "Sporting Goods Stores and Bicycle Shops." Here, the use of the word "shop" in "bicycle shops" is probably influenced by the industrial use of the word "shop." As others have noted, factories and places where machinery is repaired are called "shops." "Sorry, I can't come tomorrow, my car is in the shop." In this sense, giant factories are called shops. A factory where the workers are unionized is "a union shop." Data processing and computer professionals might refer to a company that uses exclusively IBM data processing equipment as being "an IBM shop."
It has gotten rare, but when I was a kid there were a certain number of little gift or souvenir stores with names like "Ye Olde Clock and Gift Shoppe," where the spellings "Ye," "Olde," and "Shoppe" are supposed to go along with some vague image of an idealized Merrie Olde England. I don't know if you are familiar with the "cottage" paintings of the late Thomas Kinkade, Painter of Light®, but that's the idea... "Shoppe" is supposed to suggest gaslight, cobbled streets, thatched roofs, hansom cabs, and Robin Hood... or something like that.