Schools after grade 9 need your budget.
All schools of grades 10 to 12 (koukou)need entrance-examinations(nyuu-shi).
The way of entrance exams depends on the administrative units.
But there are common facts:
1-1, your child has the opportunity to have both entrance exams, public koukou, and private koukou.
1-2, your child can choose the "recommendation (suisen)"entrance exam. The dates of suisen are before 1-1 type exams (ippan=general). If s/he passed suisen, s/he must enroll in the school.
2-1, the date of payment to the private schools is before the public schools'.
2-2, expenses for an entrance exam and enrollment to koukou(grade 10 to 12) are:
Examination fee (per school your child will take)
Tuition fee (per month)
Uniform and textbooks
2-3, You must be careful that if you hear "scholarship (shou-gaku-kin)" in Japan, sometimes they're "student loan". The majority of local administrations have school subsidies (koukou-hojo-kin).
3, vocational high school students can go to university after graduation.
4, you must check schools carefully that won't give your child "certification of grade 12" because there're many schools, institutions, facilities that make you confused anywhere.