What is the moral of this story or the lesson to be learned?

The major moral of this story is that a person must keep an open
mind.  Things are not necessarily as they seem and people who do not keep an open mind expose
themselves to danger.

This can be seen most clearly in the
interactions between Harry and his friends and the various animals (real animals or the the
animal versions of animagi) in the story.  The friends consistently misinterpret the intentions
of the animals in the story.  For example, Harry and Ron spend much of the story being angry at
Crookshanks for trying to kill Scabbers all the time.  As another example, Harry is terrified by
the big dog.  Relatedly, everyone is afraid of Sirius Black.  All of these perceptions turn out
to be wrong.

Because of their inaccurate perceptions, the friends
get themselves in trouble at the end of the book.  They also inadvertently help Voldemort by
letting Scabbers/Pettigrew return to him.

The book is all about
thinking you know something when you really are wrong.  It is about how we endanger ourselves
when we are so sure of something that we do not allow ourselves to consider that we might be
wrong.