If you're certain the ancestors actually were Native Americans and know the name of the tribe, you can Google for that tribe's headquarters and they'll tell you where to find records and such (like the Indian Rolls). In many cases, you won't have to travel to a location to do onsite research, but there may be a charge to have someone else look up the records for you.
That said, if you don't know **for sure** the ancestors were Native American...i.e. it's only a rumor passed down in your family...then you'd best look for them first in the Federal censuses (from the 1850 census through 1900) in whatever state they lived in. If you find them in the censuses, then they probably *weren't* Native Americans because Indians were enumerated in a different type of record (such as the Indian Rolls).
Good luck! ;D