You do not need to clean your breast or nipple in a special way before or after feeding your baby. Rinsing your breast with plain water while having your daily bath or shower is all that is needed. Avoid anything that dries out or otherwise damages the skin. This includes the use of soap or disinfectants on the nipple and areola as this can dry your skin and may contribute to sore nipples.
You may express a few drops of milk after feeding, massage it gently into the skin and let it air dry. If your skin is especially dry or damaged you may apply a tiny amount of nipple cream to your nipple and areola. Bras and bra pads should allow the circulation of air. Any kind of plastic lining in bra pads or bras should be avoided.
Engorgement and mastitis are best avoided through proper breastfeeding management (frequent feeding on demand and draining each breast fully) but sore and tender breasts can be soothed with compresses and ice packs
Breastfeeding is not meant to hurt. If you experience nipple pain or soreness ask your lactation specialist.