上学

Both phrases can be correct, but they point at different ideas. 去学校 focuses on the building or location. 去上学 focuses on your role as a student and the school routine. When the person is a student going to class, 去上学 is the natural choice. When someone is just visiting the campus, 去学校 is fine. The old man in the story is sending the boy to class, so 上学 is exact.

For university, Chinese uses 上大学, not 上学大学. 上学 already carries the idea of attending school. So you do not add another school word after it. If you want to mention the place, put it before 上学 with 在. 我在大学上学 is fine. 我上大学 is the normal way to say I attend university. 我上学 alone often means I am a school student or I go to school.

If you mean one class at a set time, use 上课. 我九点上课 means I have class at nine. 我九点上学 sounds like your school day starts at nine. 上学 describes the general act of going to school or being a student. It does not mean one lesson. A university student often says 我去上课, not 我去上学. Use 上学 for the whole school routine or student status.