Speaking as an American, grew up in New York, most of my life in Ohio, currently in Michigan:
I don't think I've ever heard an American say, "I am going to the shops" or "... to a shop".
If you just want to be general, you can say, "I am going shopping" or "I am going to the store."
"Shopping" can mean browsing around looking for anything that strikes your interest, or it can mean that you are looking for specific items. I guess it would be unlikely for someone to say "I am going shopping" if their intent is to buy just one or two specific items, like run to the hardware store, get a can of paint, and come home. But it wouldn't be jarring if someone used in that way, just unusual.
As nxx says, yes, a "store" is a place where goods are stocked for retail sale, while a "shop" is a place where services are performed. Thus you get your shoes repaired at the "cobbler's shop", NOT the "cobbler's store". But you buy new shoes at a "shoe store", NOT a "shoe shop". You get your car repaired at an "auto repair shop", NOT an "auto repair store". But if you're fixing the car yourself, you buy the parts at an "auto parts store", NOT an "auto parts shop".
Some cases are hazy. Like you get meat at a "butcher shop", NOT a "butcher store". The butcher cuts and prepares the meat, so he is performing a service, though I'd think the main point is that he is stocking and selling the meat.
One business may perform multiple functions. Like in my cobbler shop / shoe store example, some shoe stores also provide repair services and some cobblers also sell shoes. So there's some ambiguity what you would call them. But that's no different from the problem of what to call a store that sells different, unrelated products. Like most drug stores in America today also sell many other products, typically snack foods and grooming products, sometimes other things. But we still call them "drug stores", not "drug and snack food stores". I guess you pick the main thing they do. If they do enough different things, they become a "department store".
Oh, sometimes a store will call itself a "shop" or "shoppe" because Americans are aware of the UK usage and they want to give it a slightly exotic flair. Like when I lived in Ohio there was a shopping mall that was officially named, "The Shoppes at Fairfield Commons". As it was in the town of Fairborn, in real life everyone called it "the Fairborn mall", but they tried.
To the best of my knowledge, what I've said here applies across the U.S. I haven't noticed any regional variation. I understand it is different from the U.K. No idea about other English-speaking countries.