Do they mean the same thing and are grammatically correct?
1) I set about going to school and then I attended the University.
2) My scholastic career started off as a student at school and ended up at university.
3) She set about telling stories in the market place.
4) She started (off) telling stories in the market place.
#1 seems odd to me.
1 sounds to me ridiculous. When you 'set about' something you take the first steps in what you recognize as a long process, but which you think ought to be attempted: 'On leaving [high-]school, he set about the arduous seven-year training to become a doctor of medicinel'. If someone who lives in London starts walking to a school in Edinburgh, he might be said to 'set about going' (i.e. it's pretty improbable).
To me, and I suspect to other users of Br Eng, 'setting about' a task implies rather more than just 'starting off'.
OK, not necessarily long, but considerable. That definition is incomplete, and the example is not a good one - as there is not enough context to show that 'the cleaning' was more than a desultory flick of a duster; but 'before lunchtime' does imply that the job took quite a while.
I'm sorry but I have not understand it yet.
What's the difference between 1) I started the cleaning AND 2) I set about the cleaning ??
That is, how can I use them without making mistakes?
Say, "I started cleaning". It's always correct.
Only use "I set about cleaning" if you want to give the impression that the cleaning may take some time, or involve some effort.