Hi David,Looks like you're just beginning your search, welcome to the world of genealogy.By now you probably are aware that Thompson is very common surname, so researchers like myself need at least a teensy bit of information to be able to give you some help. (^_^)
I am assuming these people are deceased, so you cannot ask them the questions.So, lets begin with your father.When and where was he born -- state and county?That will put his parents in a time frame, and a location for record searching.Did your father have any siblings?If so, where was he in the birth order, oldest, youngest etc?This too will help with dates on his parents.
Names (even nicknames), dates (even approximate ones) and locations are very important.Most vital records are generated at the county level, so names of counties is helpful.If you know the name of a town, then here is a site that will help you locate the county. http://www.getty.edu/research/tools/vocabulary/tgn/index.htmlhttp://www.getty.edu/research/tools/vocabulary/tgn/index.html
Talk to family members who might know things like, what does T.D. stand for in your grandfather's name.Some of your relatives might remember little tidbits that will be helpful down the line.
Talk to everyone, even those who married into your family.The husband of my grandpa's cousin remembered first name of their grandfather, and his wife's maiden name.Grandpa and his cousins didn't know that, they just knew Grandma and Grandpa Walker.I was totally surprised.None of them were doing genealogy, either.I just lucked out, because talk about common surname. Have fun, and Happy Hunting.