Colin,
If I were looking, I would start with the list and check the various states to see where each individual Robert enlisted. You can also check the muster record of each Robert and that might tell you where he was from which might or might not be the same as the county/town he enlisted in.
As for his parentage, perhaps Ancestory.com can help. They have pretty good records. Also if you know the birthdates and you should be able to find that out on the muster record, you can check census records and that would give you his mother and father.
It takes a while but if you persist you can find out a lot of info.
Illinois has a lot of records of the men from Illinois and I am sure most other states have thiers also.
Frank