ピアノ教室って英語でなんて言うの？

A place where you learn to play a piano could be a 'piano school' or 'music school'. However, many tutors teach lessons privately from home and they may well refer to: 'the piano room' or 'the music room' in their home.
"Please go through to the music room."

If you are studying with a private teacher, you would say "piano class" or "piano lessons." But if you go to a place where they teach many different instruments you would say "music school."
ex: I am going to my piano class.
I take piano lessons in the afternoon.
I go to a music school for piano classes.

When you go to take piano lessons we call the place you go to a 'music school', 'piano school', orwe also say we go to 'piano lessons'. The actual lesson name can be used after the verb 'to go to'.
For example, "I'm going to piano lessons on Thursday.'

If you would like to learn how to play the piano, it helps to go for classes with a piano teacher.
"I am off to piano class, I have an exam today."
"We have a piano rehearsal tomorrow at 9 am."

Music school
This is what the school is called where students go to learn
how to play all kinds of instruments including the piano. In
other words, it is a general music school.

Piano Institute
An institute is the name given to an organization or
establishment where any type of education/ learning takes
place.

Private piano tuition
Many piano teachers enjoy teaching from the comfort
of their homes and this is called private piano tuition.
The student is coached/tutored alone (privately).

Both words are used to describe a place where music is taught including how to play the piano.