The easiest place to start on the internet is Civil War Soldiers and Sailors System at:
Francis Kirk (First_Last)
Regiment Name 1 Cherokee Mounted Volunteers.
Soldier's Rank_In Private
Soldier's Rank_Out Private
Film Number M818 roll 14
What can you tell us about Francis Kirk? There is also a George Kirk in this regiment. Are they related?
By the way, at the top of this page is a link to order service records and pension records through History-Sites.com.