You can take advantage of Ancestry.com free trial and search there -- the site has a very good selection of indexes to service records and pensions records as well as the surviving 1890 US Census Veterans Schedules for States alphabetically after Kentucky.
You can look up your man in the National Park Service's Soldiers and Sailors index:
If you find him, note the Micropublication number and roll number for the entry or entries.Then you can ask your local library to borrow the microfilm by inter-library loan from the National Archives, so you can look at and make photocopies of his service record.
The site www.footnote.com has been putting up Civil War Pension Files images on its site.It, too, has a free trial period which does not allow copying, but since getting a copy of the complete file from the National Archives now costs at least $100 it may be worthwhile to subscibe to the site for a month.But do a search first -- it has not finished putting the images up.
If you find a pension index listing on ancestry.com, note the Certificate number for the soldier and that for his widow if she applied, as well as exact spelling as given in the card.Then you can get your library to borrow the pension application file microfilm from the National Archives on microfilm - these are arranged alphabetically by soldier and there may be 100 or more pages in the file.These files can be very genealogically rich, giving lists of children, residence locations over the years, affidavits from friends and relatives.
This message board is for the US Revolutionary War (1775-1783), not the US Civil War (1861-1865).Any further queries should be posted under your surname, on the Civil War message board, or on the message board for the County in which your man lived.