If your underarms have become a bit red, bumpy and irritated, look at your antiperspirant before anything else. Antiperspirants do just as the name implies: They block your sweat. Most antiperspirants contain aluminum salt to block your pores and keep you dry, but that ingredient may also irritate your skin. The more aluminum salt your antiperspirant contains, the higher the chance it will cause irritation (and possibly damage your shirt, too).
Antiperspirants that contain alcohol may also be too drying for some people's skin, which can lead to irritation, as can any ingredient that ends in "-paraben" (such as methylparaben).