What are some major conflicts throughout the main characters in "The Wednesday Wars"?
I would say that the main conflict throughout the novel is between Holling and Mrs. Baker. I would not say that they are protagonist and antagonist through the entire novel, though. The conflict is with their changing relationship. Let me be clear on that. Their relationship is always teacher/student. Never romantic in any way. When the novel begins, Holling is forced to be present in Mrs. Baker's classroom on Wednesday afternoons. He's not happy about it, and Mrs. Baker isn't happy about it either. In fact, she has him do grammatical sentence diagramming and chalkboard cleaning. Yeah, not fun. But as the year goes on, Mrs. Baker starts reading Shakespeare with Holling. He learns to love it. His passion allows him to get the courage to sign up to act in a Shakespeare play. He decides to join the track team, and Mrs. Baker (a former Olympic runner) helps him train. In return, Holling helps Mrs. Baker find ways to develop a better rapport with her other students. Their relationship never stops growing throughout the novel.
A second major conflict is between Holling's sister (Heather) and her parents. It's not a central focus to the book, but it is a big conflict. Holling's father is incredibly ambitious. He's stubborn, too and holds high standards. Holling's mother is completely passive and goes along with whatever her husband says. The book is taking place during the Vietnam war, so social and political opinions are all over the place, and Heather is becoming rebellious against her parents. It gets to the point where she runs away from home. Heather gets stranded, but Mr. and Mrs. Hoodhood do nothing. It's Holling that sends her money for the bus ticket home.
A third conflict is Mrs. Baker's own internal struggle. Her husband is fighting in Vietnam. She has little to zero news about him, and the news media is continually showing bad news about the war. She's worried about him constantly, and she internalizes most of it. She's having a fairly good year at school with her students, but she is a mess on the inside.