Engoeged breast is a quite common problem which many mothers encounter during initial days of motherhood. It’s a very painful condition. As the milk comes in for the first time, the vessels surrounding the lymph nodes are flushed with blood. Engorged breasts are hard, are hot, can hurt, and-as you pointed out- can make it difficult for a newborn to latch on because the areola is too firm. There are several ways to relieve engorged breasts:

• Take a very warm (but not unbearably hot) shower. Stand with your back to the shower, and gently massage out the milk by hand. Express until your breasts are soft enough to allow the baby to latch on and suckle. Your breasts will still feel huge, but more pliant
• Warm a couple of hand towels in the dryer or microwave. (They're ready when they're hot enough to barely handle, like the heated towels that customers use at Japanese restaurants.) Lay the warm towels on your breasts. The heat probably will inspire some letdown, which will relieve the pressure, and the towels will absorb the milk.
• Use a breast pump (electric pumps are better than hand pumps) to empty some milk from your breasts. This is easier than taking a shower if you're still recovering from a c-section, and you can save the milk in the freezer to use later.
• Let the baby nurse as often as she wants. This will help your milk supply stabilize more quickly and reduce the engorgement
• A well-fitted, supportive nursing bra makes some women feel better. Others prefer to go bra-less during engorgement.
• Gentle breast massage and relaxation techniques may help improve milk flow and reduce engorgement.
• Fever higher than 101° F or severe pain may signal a breast infection. Call your medical care provider for advice if this occurs. 

How to prevent Engoeged breast:

• Begin breastfeeding as soon as possible after the birth, to give the baby time to learn to breastfeed before the breasts become full and firm.
• Unless medically indicated, avoid early use of bottles and pacifiers while the baby is learning to breastfeed.
• Once the milk comes in, breastfeed at least 8 times in 24 hours to prevent over fullness.
• Ask for help from the hospital lactation consultant so that latch-on problems are solved as soon as possible.
• Any time a feeding is missed, use hand expression or a breast pump to remove the milk.
• Always wean gradually