Getting a good latch is indeed very important, and if you suspect your baby is not latching properly now is a very good time to work on it, before it becomes a hard-to-break-habit for your child to latch in a suboptimal way.
First, the size of the areola, "the brown part of the breast", varies between women. Some women have very small areolas, others have very big ones. Sometimes, the whole areola won't fit inside the mouth of the baby, and that's ok too. What is important is that the baby gets a big mouth-full of breast, and doesn't end up just nibbling at the nipple (this will mean a lot of pain for the mother, and not so much milk for the child).
Something a lot of new mothers don't know is that you need to use a little bit of gentle preassure in the beginning to get the baby on your breast before she closes her mouth. Gently rub your nipple against her upper lip, and she'll open her mouth. As she opens, quickly press your childs mouth against your breast, aiming the nipple upwards to her pallet. If your breasts are engorged or your nipples flat or inverted, you might want to create a "breast sandwich" by compressing the breast just behind the edges of the areola using the thumb and index finger of the hand that is not holding the baby, to make it easier for the baby to get enough breast into her mouth. Apply the gentle preassue on the baby's neck and shoulders, but be careful to not push on the back of your baby's head (most babies will react badly to this). If your baby does not latch on well, insert your pinky into the corner of her mouth to break the suction, remove your nipple from her mouth, and start again.
It can be very hard to learn how to get a good latch simply from reading about it, and having someone show you is always the best way. If you don't have access to someone who can teach you, you can always rely on videos online. I learnt a lot from watching this video about how to latch a baby on.
Regarding the supplementation, I already offered an explanation in another question as to why this is not a very good idea if you wish to build a good milk supply for your baby.