Study can be used as follows:

(1) study a subject
(2) study for an examination or qualification
(3) study hard, intensively, a lot etc.
(4) study at a school, university etc - but this is rather formal and not very common. And in some cases mildly ridiculous:
My four-year-old son is studying at kindergarten

Note that the word 'study' suggests a serious, mature activity. It is not really suitable for describing the gentler, less intensive activities done in kindergartens and primary schools. (Although some Hong Kong primary schools might like to think otherwise!)

Examples;
He studied Medicine at university.
She is studying for her Form 5 exams.
If you want to be a lawyer, you'll have to study hard.
Michael Smith studied at Westminster School and Goldsmith's College, London.
(This is rather formal. It might be used for printed biographical information such as that given for an author on the cover of a book etc.)


My younger brother goes to / is at kindergarten.
She's doing Form 5 this year.
Which school did you go to?

Education words that do not collocate with study.

(1)course
You cannot study a course. You take or do a course, or more foramlly, attend a course (formal written style).

(2)educational institutions
You cannot study a school You can (rather formally) study at a school. But in spoken English you would normally go to a school, and in formal written style (job application letters etc) attend a school.