Can we say
I have been at New York, Chicago, etc. (I have been to New York.)
or this is only used for a specific places
I have been at The Metropolitan Museum of Art. It was great. (I visited it and came back to my home)
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"I have been to" would always be correct, whether it's a city, a museum, or a friend's house.
"I was at" a museum would be correct, or "I was in" a city.
There is a bit of nuance here: If someone asked, "Where have you been all day?" one might say, "I've been at the Met. It was great!" However, if someone asked, "Have you ever been to the Metropolitan Museum?" or "Have you been to any good museums?" then one would say, "I've been to the Met. It was great!"
Side note: "at" is often used incorrectly by American English-speakers, e.g., "Where are you at?" or "Where's that at?" In both examples, the "at" should be omitted.
Using the word "at" in this example implies that you have been "at" a particular place in or near the city. This place can usually be determined from the context. For example:
When speaking about a sports league that has teams in multiple cities, the place is usually the "home stadium" of a team from that city. For example, "The Seahawks played at Cincinnati on Sunday" means that the Seahawks (a team in the National Football League) played at Paul Brown Stadium (the home of the Cincinnati Bengals) on Sunday.
When talking about a person's academic career, the place is usually a school or university that shares its name with the city. For example, "J.R.R. Tolkien spent many years at Oxford."
Other places might include a major plaza, a train station, an airport, or the gates of a walled city.