Cotton shirts don't need dry-cleaning. Just regular laundering. And, in fact, the harsh chemicals in drycleaning solution are probably worse for the fabric.
To get the longest lifetime, its probably better to home-launder them. Commercial washers, dryers, and irons can be a bit tough on fabrics.
For home laundering follow this process:
1) Don't let stains "set" in fabric. Treat any accidental spills ASAP. Pre-soak troublesome spots.
2) Wash in the coldest possible water. White shirts can get by with a "warm" setting. Colors, especially bright ones, need cold water. You can use a non-chlorine bleach (Borax, etc.) to brighten fabrics.
3) Iron and starch your shirts. Starch sprayed on the fabric actually creates a temporary stain-guard, so it is especially good at protecting collars from unsightly stains.
4) Repair lost or loose buttons promptly. I taught myself basic needle-and-thread while in college, and my ability to replace lost or loose buttons and do small mends has amazed several girlfriends.
Most importantly, treat your shirts properly. Don't fall asleep in a good shirt. Don't mow the lawn, clean the kitchen, or change the oil in them. Hang them on decent wooden hangers, and leave an inch or two "breathing" space between them in your closet. "Special Occasion" garments, such as tuxedo shirts, can be folded (like they come from the store) and wrapped in tissue paper. (This prevents them from getting a coating of dust sitting in the closet for six months.)