I would start by looking for obituaries for your Dad's siblings (if he had any) and for your grandparents. You can sometimes find obituary indexes at the local public library for the city they died in. You can find some cemetery records, which may give you an exact date of death at www.findagrave.org
Once you have names of living relatives, you can try searching for them at a site like www.switchboard.com
If you tell us what cities and states your family died in, we can give more specific suggestions for obit lookups.