Rachel, if one of your parents were Indian you would be 1/2, if one of your grandparents 1/4, if two 1/2, if three, 3/4. Do you see how unusual being 1/3 Indian would be?
That said, you need to start with your parents and work back. Starting somewhere in the past and working forward will just confuse the issue (many Indians in the southern U.S. have Scots names). Keep good records and interview all the living relatives you can. Then start checking census records.
If you find a link to any tribe, check that tribe's HQ for information. There are "rolls" of people recognized as members of the various tribes. In the case of the Cherokees, these are known as the "Dawes Rolls".
Mike Shaw in Texas