What is the first major hurdle charlie overcomes? How is it resolved?
I am not sure if you mean in his life or after he has the operation so I will answer both.
He goes to night school to learn to read. He is highly motivated - this is a big step for him. He wanted to be considered for the experiment so he had to keep a journal and this was difficult for him. Initially, the doctors considered his IQ too low to be of use to them but his persistence and the support of Alice his teacher caused the doctors to reconsider.
After the operation, his first big hurdle is to do the maze quicker than the mouse. Initially before the operation, the mouse beat him. After the operation, he does the maze again and again until he beats Algernon, the mouse. This would be considered his first hurdle.