The most central problem is that Doodle is born disabled. This is a conflict of man vs. society. In the time period of the story, the disabled had very little chance for a normal life. Doodle will grow up needing medical care that isn't available, and will face being an adult without ever being able to get a job and support himself.
Other social conflicts that are mentioned include drought and crop failure. As the boys live in an agricultural community, these problems mean a lack of income for local families, less money to pay for food and for doctors, which directly affects Doodle and his brother.
The narrator has a few internal struggles. He does not like having a disabled brother, and feels ashamed of Doodle. This is a problem because it forces him to push Doodle to improve. Although this causes some good changes, it also leads to bullying and eventually to Doodle's death.
Doodle himself has an internal struggle. Doodle wants to please his brother. He feels that is a failure to his family and he wants to be accepted, as most people do. He allows his brother to push him and takes his failures very personally. This also helps to lead to his death.