Most dress shirts, especially if they are cotton, can be washed in a washing machine and then dried on a low-heat setting or hung out to air dry. While dry cleaning is an option, it is not always the most effective.
The first thing to do when figuring out how to launder a dress shirt is to always check the tag on the inside of the shirt. For many dress shirts, this will be on one of the side seams towards the bottom. This will determine the washing temperature and which type of detergent to use.
Before washing the shirt, be sure to unbutton all the buttons and remove the collar stays. This will protect the buttons and keep the stays from warping. If possible, pre-treat any sweat stains or other stains before washing the shirt. This can be done by rubbing a small amount of detergent or dish soap on the spot or using a stain remover pen.
For darker shirts, wash them in cold water and use the delicate setting. For lighter shirts, use the delicate setting, but wash them with hot or warm water. Remove the shirts from the washing machine as soon as possible, and hang them to air dry. Avoid using the dryer if possible, as high heat can shrink the fabric and alter the color. Once the shirt is mostly dry, iron out the wrinkles and hang the shirt on a thick wooden hanger to help it keep its shape.