As children grow up, they will take classes in school that will have them start questioning their family tree. Social studies and history classes are the biggest catalysts, as the subjects broached include wars. Whether you're wondering if your dad was in the Vietnam or Korean wars, or if he had a part in either war in Iraq or the war in Afghanistan, it shouldn't take you too long to find your answer.
Ask your mother about your father. However, if your mom is not around any more to answer your questions, you may have to search further.
Ask your dad's siblings and your grandparents if he was a veteran. They will remember if your dad was enlisted in the military. If you're not in contact with your dad's family, but you are with your mom's family, ask them.
Request veterans' service records from the government. Visit the National Archives website and search for service records. Next of kin should be able to request their parents' DD-214s (discharge records) for free. This search will tell you if your dad was a veteran.