School names can cause problems for both English and Czechs, because the system is so different. In England, children start school at four years old. From 4–11 they go to primary school (elementary school in the US), from 11–18 they go to secondary school (high school), and after this, they may go to university (college).
Basic school as a term doesn’t exist and if you say it, it suggests a school that is very basic, without modern equipment or is undeveloped. A grammar school, historically, is a school where students are admitted on the basis of ability, usual decided in the form of a test. Other schools in the UK are called comprehensive schools.
Anyway, here is the same sentence from a Brit and an American:
Brit: I went to primary school but I didn’t like it. My secondary school was much better, although I didn’t get good enough marks to get into university.
American: I went to elementary school but I didn’t like it. My high school was much better, although I didn’t get good enough grades to get into college.