I have an ancestor (William Miller) who on his son's baptism record (Woolwich, London, 1813) is listed as a 'Private in the Driver Corps'.
How can I get more information?
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My stock answer to the question "How can I get more information?" is usually any way you can -- especially in the case of someone with a common surname like Miller. Your problem is not simply finding more information; you also want to know, once you find a record, that the record you're looking at belongs to your William Miller, and not one of the many other William Millers who might have also been in service.
Looking at the scant information we have from this certificate, I note the following:
If, starting with this big set of assumptions, I go to the British Army Service Records 1760-1915 record set on Find My Past, and enter "William Miller", I get 2320 results. If I narrow this by typing "artillery" into the keyword search field (experience has shown this is more reliable than putting "Royal Artillery" into the field for 'Regiment'), that narrows the result set down to 60.
You did not say what kind of birth record this information comes from. I deduce that it isn't a regimental register of births, or you wouldn't be asking this question in the first place.
Now -- assuming my guesses are all correct -- you still need some way to distinguish your William Miller from the other 59 men of the same name (assuming, of course, that the record set is complete which is often not the case).
In my case, I was able to determine I had the right person because his marriage was recorded in his service record. I also knew where his wife's relatives lived, and had other clues from gathering "possibles" from the census of what some of his his street addresses might be.
To get more general information about the RA, you could explore:
For more information about Service Records see:
I agree with the general suggestions made by @JanMurphy, and would add this specific finding, which may or may not be your William Miller.
A search of Ancestry.co.uk's UK, Military Campaign Medal and Award Rolls, 1793-1949 reveals a William Miller who fought in the Battle of Toulouse in 1814 during the Napoleonic Wars. He was a Private in the Artillery Drivers, under Captain Brandreth.
I do not have access to the records myself, but I advise you to investigate these two service records in FindMyPast's British Army Service Records 1760-1915.