One conflict that Jonas struggles with is how to handle all of his newly acquired feelings and knowledge. As the receiver of knowledge, it's Jonas's job to be given all of society's history and emotions. That way the information will be kept in tact for when it is needed, but none of the actual population will have access to it. Jonas understands the purpose of the rule, but doesn't necessarily agree with it. Plus he's completely unprepared for the (at times) painful process of his job.
Another conflict is what to do about his brother, Gabriel. Jonas discovers that his brother is going to be released. Jonas has to decide whether or not to let it happen or find a way to rescue Gabriel.
A third conflict that Jonas has is what to do about the Sameness. He might disagree with the concept of the Sameness, but deciding to actually do something about it is another matter. It's an internal conflict that he has, and Jonas eventually begins to discuss his feelings with the Giver.