There is a surprisingly simple answer to your great question: Each month we see different stars, so to answer for a specific night of the year at your specific location on Earth depending on your eyesight and how bright the city lights around you are might be hard to pin down.
However there is a magnitude scale that astronomers use to describe how bright objects in the sky are. Without getting into details on how that works, the average human being can see down to about magnitude 6 ( brighter objects have lower magnitude until you reach zero.
Then even brighter objects use negative numbers for their magnitudes which is a bit weird, isn't it?)
Therefore, all you have to do is get a map of the whole sky showing anything brighter than magnitude 6 and count the stars. If you are fortunate to be away from bright city lights and you can see the Milky Way, you would be really surprised to know that you can only see about 6,000 stars.
That's the number of stars above magnitude 6 in the entire sky.