Hello Rob,

A decent public library with computer access to Ancestry.com or microfilmed census records is a good place to start. Do you know who this man's parents were? His wife? Where did he live postwar? Year of birth?
Without these clues you will probably be overwhelmed by the information about "James Bird/Byrd" that may be posted.

James E. Byrd 34th Virginia Infantry

James E. Byrd 40th Virginia Cavalry

James E. Byrd 24th Virginia Cavalry

James E. Byrd 52nd Virginia Infantry

James Byrd 14th Virginia Infantry

James Byrd 59th Virginia Militia

James Bird 31st Virginia Infantry

James Bird 162nd Virginia Militia