Hi,
You can try sending for your grandfather's military records. But, there was a terrible fire in St. Louis and many of the original WWII army records were lost. So, what you get is what they have been able to reconstruct. If your parent (son or daughter of the grandfather) is still alive, it will be easier for them to order the record.
http://www.archives.gov/research/index.html
See military records on this site.
Ricbru is telling you to start with the known and work backwards. The copy of the grandfathers death certificate is one document. GEt a copy of your grandmothers death certifcate. Get copies of obituaries. These will give you information to work further back with.
You can also file a FOIA with INS to try and get informaiton on your grandmother.
http://uscis.gov/graphics/formsfee/forms/g-639.htm
Because you have so few facts to work with you have to gather them and work backwards. Does the Uncle, who was possibly born in Italy, still live? If so call and ask questions. If he is deceased, perhaps one of his children have his birht record or know where in Italy he was born. GEt his obituary. You need to identify the town in Italy they came from in order to do research in Italy.