You're right, it means "after three lessons." Good dictionaries should have this meaning, but since "on" is one of those words that are used for a million different things in English, you may have to scroll down a bit to find it.
Cambridge Dictionary says:
on adverb (MOVING FORWARD)
B2 continuing forward in time or space:
They never spoke to each other from that day on (= after that day).
What are you doing later on?
When you're done with it, would you pass it on to Paul?
UK Move on, please, and let the ambulance through.
UK You cycle on and I'll meet you there.
For what it's worth, as an American I don't agree that the last two examples are UK-specific. These usages are found in Am.E too.