Students are called 生徒 seito at school and 学生 gakusei at university; 学生 is also the general word for “student”.
Short answer: Japanese schooling runs 6 years of 小学校 (primary), 3 of 中学校 (junior high), 3 of 高校 (high school) and usually 4 of 大学 (university). The year starts in April. For a learner, the most useful school words are the classroom phrases: 分かりません (I don’t understand), もう一度お願いします (once more, please) and 質問があります (I have a question).
School subjects
国語 (“national language”) is Japanese as a school subject.
Classroom phrases for learners
Useful in any Japanese class, and in any conversation where you get lost.
School life
Clubs, uniforms and exams shape Japanese school life.
Clubs
Uniform
Exams
Homework
University life
Words for university courses and campus life.
Stationery
Japanese stationery is famous, and the words are useful in any class.