There are many options to select from. You can either buy a military issue shirt in white muslin or gray wool, or you can go with a civilian shirt in any period fabric you choose. Most of the time your shirt will be completely covered up by your jacket. But when you are lounging around camp on a hot day, you'll likely be in your shirtsleeves - so you'll probably want it to make an impression. The shirt above is made of pillow ticking.
Some fabrics are much more vibrant - bright paisleys, checks, plaids or solid colors.
Just about any period shirt will work for either a Federal or Confederate impression. In addition to machine-stiched shirts, it is also possible to purchase shirts with handsewn seams.