Are both "see you at the lesson" and "see you in the lesson" correct?
Solution 1:
In American usage, both of these phrases are correct grammar, but they don't mean the same thing, and at is probably the intended meaning. At merely implies you and the teacher will be in the same place. In would be acceptable if your lesson were a performance, which in a classroom setting it might be. You might find this easier if we contrast "see you at the movie" with "see you in the movie".
Having said that, prepositions can behave rather arbitrarily and non-American usage may differ.