This phase III trial studies the side effects and how well risk-based therapy works in treating younger patients with newly diagnosed liver cancer. Surgery, chemotherapy drugs (cancer fighting medicines), and when necessary, liver transplant, are the main current treatments for hepatoblastoma. The stage of the cancer is one factor used to decide the best treatment. Treating patients according to the risk group they are in may help get rid of the cancer, keep it from coming back, and decrease the side effects of chemotherapy.