"Barnet går på skolen."
Translation:The child goes to school.
Correct. "Å gå på skolen" refers to the act of going to school as in being in school, not to the process of getting there.
Is the definite form always necessary in cases like this? Could, "barnet går på skole" also work here? And if so, would the meaning change?
No, but it's more common in most situations.
"Å gå på skole" is often preferred when going to school is used in a very general sense, in sentences like "All children have the right to go to school".
I'm noticing that, in listening exercises, I can't tell the difference between barnet and barna. Anyone else have this problem?