take a lesson - a paid lesson for example. do a lesson - to begin / to work on a lesson.

'Do' is a multi-purpose verb. The verb 'do' can replace many different verbs. It is extremely flexible, and extremely useful. But there is a problem, because it's not always 100% clear what it means! 'Take' is also a very flexible verb with many different meanings that can replace other verbs. Not quite as many as 'do', but again it's not always possible to know what it means. In both cases you need to use the context to understand the meaning and/or ask questions to clarify the meaning. Let me give you an example which shows the verb that I am replacing: Student: Hello. What type of lesson do you do (offer)? Teacher: I do (teach) online group and one to one lessons. Student: Can I take (have) a lesson on Friday at 10 o'clock? Teacher: I'm sorry I am taking (covering) a lesson for a colleague at that time. But note that there are many possible alternatives that could replace the words in brackets.

When you say "Take a lesson" it means you're learning a lesson from someone, (a teacher, friend etc.) When you say "Do a lesson" it still means you're learning a lesson, but not necessarily with the help of someone else.