The first rule of genealogy is "start with what you know." So, you've got your start. If your grandparents are still alive and coherent, ask them for further info. If they're not, start using the web. www.familysearch.org is free. There is a lot of info there.
The Church of Jesus Christ of Latter-day Saints has small family history libraries in many of their church buildings around the world. They are staffed by volunteers who would be glad to help you get started and can look at what you've got and give you advice of where to go next. You do not have to be a member of the church to use the libraries, as a matter of fact, the last one I went to said that 95% of the users were not members of the LDS church.
Specifically, since it sounds like contacting your grandparents is not an option, but you do know their birthdays, I'd check the Social Security Death Index. (SSDI) (Available either on family search or ancestry's websites) You can match up your grand parents names with birth/death information and, if you find a match, you can order their application information. This will show their parents and other information.
Another possibility is, if your grandparents are old enough to have been children on one of the available census (they are made public after 70 years) then you may be able to find them listed as a child -- try searching first in the area where your parents grew up if you don't have any better location information for your grandparents. A hit on the SSDI may also give you some possible locality information.
Also, 2 helpful genealogical programs. The LDS church offerss its genealogical storage program (PAF) free and it's downloadable. There are links from familysearch.org. Get it, and enter the information you have. Then, get a program called GenSmarts. It works with the PAF, analizes the information you have and suggests where to go next. You can get a free trial at their website: www.gensmarts.com
Warning: Genealogy is addictive. Have fun!