I usually say 7-11 even though they all left my state over twenty years ago. 7-11 was the original mini-mart in my childhood town.
Or I’ll say the Exxon because its two blocks away and that’s where I buy gas and get milk, bread, candy etc.
other terms I can think of
Mini-mart, Gas Station, Convenience Store, or maybe a specific local store name?
I’m sure there’s store names I’m not familiar with.
When I lived in England we used to call it “the corner shop” which often was shortened to “the corner”.
Here in Italy you go to the alimentari.
I might say Diamond Shamrock, since it’s the closest place. Although I’m in the middle of Circle K territory, so that’s what most people in my area are likely to say.
I always say “Kwik-E Mart” because of The Simpsons.
They are all of those things.
7-11 is ubiquitous, and in my area there are some neighborhoods where 7-11s are literally only a block apart, but it has still not become a generic name for all convenience stores.
There used to be 6-Twelves but those have all become something else now.
Almost all gas stations have stores now. Some even have local chain restaurants attached (Jerry’s Subs & Pizza around here).
There are corner stores and Beer & Wine Delis everywhere.
Everyone I know just calls everything by what it is specifically or just goes out to “the store.”
Gas station, because for me that’s what it is. It’s 10 minutes to “the gas station” or it’s 20 minutes to “the grocery store.”
Around here most of them are either actual 7-11s or AM-PM minimarts attached to Arco gas stations.
It is amazing how many people use the actual store name. Circle K, Exxon, Shell or whatever is closest to your house.
I can remember when milk was cheaper at the gas station. It was a product incentive to get you to stop and hopefully buy gas too. Today, milk & bread costs more at a convenience store than a grocery store. That convenience comes at a price. 
I miss the old mom & pop country stores. I bought a lot of RC Cola and moon pies there. They were the first small grocers in rural areas. Sometimes they sold sandwich meat too. But the mini-marts pretty much put them out of business.
Stop and Rob
Corner store
I call it the gas station because in our area it is rare to find a shop that sells milk, bread and candy that is not a gas station.
We used to have Convenient and Dairy Mart but the gas stations in our area sell just as much or more than they used to.
I checked off “convenience store,” but I’m likely to use the actual store’s name (mostly 7-11 and AM-PM around here). I’d never buy milk or bread there, though.
merriam-webster says convenience store was first used in 1965.
mini mart in 1981
I usually say “The Store.”
Joe
“Convenience store” is the generic term, although I would refer to the local “corner store” when I was a kid.
But when I refer to a specific one, I use the name of course. Duh.
Same here, except I checked the store’s actual name. The grocery store is close enough that I don’t have to pay the outrageous prices at a gas station/convenience store for milk and bread and such.
We have a ton of Circle K’s around here and they are known as the Circle Jerk in my family…
I just call it White Hen Pantry. You know, despite the fact that it’s no longer White Hen Pantry, and hasn’t been for yours.
In all seriousness, I call it the “convenient store” if I’m just being general about it. I know that’s horrible and wrong, but so it goes. Usually I use the name of the specific store, though.
Convenience store, but if I’m in South Jersey I’ll most likely say that I’m going to Wawa to pick up some hoagies and a jug of diet peach green tea.
The petrol station or 7-11 if that’s what it is. Other than that we’d say ‘the shop’, or ‘the dairy’ (which is a particular NZ reference), Australians would call it ‘the milk bar’.