One of the first things you may want to do is to interview your family members to see what they know about your family history. Preparing for these interviews will help you to ask the right questions, provide your interviewee with enough context to provide relevant answers, and make you better prepared to react to what the interviewee is telling you. Here are some resources that will help you think about and prepare for your interview.
Conducting Oral History Interviews: On the Record
Conducting Oral Histories with Family Members -- A guide created by UCLA Libraries.