No, you do not need to use "the", because you are not talking about one specific school, in comparison to other schools.
In this context, the child is talking about his experience of 'school life', not about the physical place, the buildings, this particular institution etc.
I (from the UK...) would not say to my daughter, "Did you have a good day at THE school?" However, I might say to a friend, "Does your son go to THE school opposite the church?" - because I'm talking about a particular school, in a particular placeand not any of the others.
Of course, you can use "the" in this example if you wish. It's not "wrong". It's just not necessary. I would say that it's superfluous.
Actually, I agree with Tula: I think most people would be much more likely to say "All the teachers at school..."