Some of the documents that will help you are a full copy of your birth certificate, a full copy of your father's birth certificate (that will give you his mother's maiden name among other things), your grandmother's death certificate.
A copy of your grandmother's obituary may give you the name of your father's siblings. If your father is deceased, his obituary may also give you the information.
If you know where your father and his siblings grew up, high school yearbooks may also help you. Many public libraries have copies of the yearbooks for the local high schools that span many years. You might have to check the yearbooks for several years.
If you have any relatives that knew the family, ask them what they know.
You might also try to find out if your grandparents had information on their engagement and/or marriage published in a newpaper. Many couples did this in the 1940's and 1950's. Frequently there was also an engagement, and or wedding picture with the article.
Birth certificates, death records, and marriage liscense records can all give you information about a persons parents and grandparents. It can be worth your while to look at as many of these documents as you can get. While they all ask for similar information, different varions of the same information may be found on diffent documents. They were filled out at the same time or even by the same people. One of my grandmother's was born in Canada. The only lace I found the name of the town where she was born was in my unlce's birth records. And I had her death certificate, and had read the original records for her marriage and my father's birth.