The easiest place to start on the internet is Civil War Soldiers and Sailors System at:

http://www.itd.nps.gov/cwss/index.html

...which shows....

Francis Kirk (First_Last)
Regiment Name 1 Cherokee Mounted Volunteers.
Side Confederate
Company I
Soldier's Rank_In Private
Soldier's Rank_Out Private
Alternate Name
Notes
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.