This tells the reader that Zach is both very ambitious and extremely courageous. He wants to get the best education possible to prepare him for the furutre he desires as a lawyer, and he knows that the education offered at the "white" school is superior to the one he would receive at the "colored" school. He is an intelligent and articulate young man and willing to work hard, and he wants to be able to challenge himself to reach his full potential.
He is also, though, extremely courageous. He is desiring to integrate a white school at a time in the South when many whites were fighting against just such integration. The most famous episode demonstrating the white hostility to such integration came in 1957 when the "Little Rock 9" faced many incidents of physical, emotional, and verbal abuse and harassment when they integrated Central High School in Little Rock, Arkansas.
Zach has already seen the reaction of some whites in his town when they feared a black woman was going to go to their theater - and he has faced the consequences for being in the wrong place at the wrong time (when his friend threw the bottle at the white men, resulting in Zach, too, was arrested). He knows the potential danger of what he is doing, but he sees it as just one step in reaching his goal of becoming a lawyer and helping to advance the cause of civil rights in the South.