Where do I go to find and order records for a Civil War Veteran?
I have the wife's and child's pension numbers.
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Like many things it will all depends on what specific piece of information you are seeking and how much effort you are willing to put forth to obtain it. For most it seems most seek Civil War Service records of simply what units they served in and which battles they participated, luckily that information is more readily available but going beyond that can involve extensive additional research if the information is even available.
US Military records, if available, can be found on various sites in different forms but the most complete collection I have found to date is on Fold3.com, an Ancestry.com company, and your Ancestry.com membership does not inherently give you access to it. Some of Fold3's collections though are available via Ancestry.com (see notes below).
Fold3 has in their collection any US Government and some state collection public records for the wars primarily from the US National Archive records as well as has many of their own commercial and crowd sourced projects (like the War of 1812 pension record preservation project currently going on and at the time of this writing is only 30% funded) and has a collection expanding and focused on military records.
What you will not find on Fold3 is records in individual state collections that have not been released to them for commercial use and are retained in state libraries or have not yet been digitized or transcribed. Some records are only available as transcribed records found in books as the original records are also lost, a lot of which seems to have been done about a hundred years ago and you will not find those on Fold3 as well.
If you are looking for simply their draft cards and general information try the free resources first and then utilize other resources you have first to determine if they meet your needs.
Detailed Service Records (if even available) for more specialized and focused research.
Keep in mind that no collection is complete and because you don't find it in one doesn't mean you won't find it in another, you may also encounter conflicts based on transcription of the records by different individuals or machines.
If you are looking for detailed records that go into enlistment date, dates of promotion, and move by move detail of his unit you are going likely have to sift through Fold3's information as well as research the unit in other sources, including state libraries, but you still may be disappointed.