If you know your ancestor fought, but not much more, here is a good place to start.
Once you know what state they fought for, try a Google search using the following terms to find records for the state: "state name" "civil war" "records" "pension" "muster" "units" "veteran"
Once you find a site to search on, search on all possible spellings of the soldiers name, both first and last; if they were recent immigrants, search on just their first name to account for "Americanization" and spelling.
Search pension records for wives, mothers and daughters names, not just for the soldier.
If they were young, remember that many young boys lied about their age in order to fight, so take the reported age with a grain of salt.
Once you find the unit or regiment they fought in, then you can search battle records for skirmishes they fought in. This then makes official government records easier to search, at least on the Union side...I have to apologize, I am from a northern family so have not had much experience dealing with Confederate records.
I would like to see a discussion devoted to research in each state, so be sure to browse the discussions for help.
Here is an all state list:
A list of Union Regiments, by state:
Here is a site with an extensive list of links to many Arkansas Civil War sites, lists, and information.
I just found this Wikipedia page...full list of all units, Union and Confederate:
I have a more definitive list of units by state on my site at www.researchonline.net and the battle list by states and years is more comprehensive than you have cited here.
I am interested in corroborating on this project here on Geni.
Just found another site with both CSA and USA that has correspondence, orders and returns, as well as all seizures of property in the south and all field operations.
Anita M. Boynton units in the state of new york http://dmna.state.ny.us/historic/reghist/civil/rosters/Infantry/