This phase II/III trial tests the safety, side effects, and best dose of the drug cabozantinib in combination with standard chemotherapy, and to compare the effect of adding cabozantinib to standard chemotherapy alone in treating patients with newly diagnosed osteosarcoma. Cabozantinib is in a class of medications called kinase inhibitors which block protein signals affecting new blood vessel formation and the ability to activate growth signaling pathways. This may help slow the growth of tumor cells. The drugs used in standard chemotherapy for this trial are methotrexate, doxorubicin, and cisplatin (MAP). Methotrexate stops cells from making DNA and may kill tumor cells. It is a type of antimetabolite. Doxorubicin is in a class of medications called anthracyclines. It works by slowing or stopping the growth of tumor cells in the body. Cisplatin is in a class of medications known as platinum-containing compounds. It works by killing, stopping or slowing the growth of tumor cells. Adding cabozantinib to standard chemotherapy may work better in treating newly diagnosed osteosarcoma.