I didn't leak at all under the end of my pregnancy, and then that was only when my nipples were physically stimulated. But i'm exclusively breastfeeding my daughter, and now you can't even tell me "hello" without my boobs spraying milk everywhere.
Breastfeeding is hard at the beginning. Your baby is so eager to suck that they'll grab onto whatever they can and just go for it, whether it's the very tip of your nipple or the side of your breast. Don't leave the hospital without speaking with at least one lactation consultant, even if you think you're doing it right. (if you're not, you're in for some serious pain very very soon.) they'll help you with different nursing positions, getting your little one to latch on properly, and teach you ways of dealing with nipple pain (because most likely, after a day or two, your nipples will feel like someone's been rubbing them with sandpaper!)
just remember to stick with it! If you correct your baby's latch every time (which will hurt, because your nipples will be sooo sensitive), the pain will fade, and they'll learn the proper way soon. I was breastfeeding .Aly comfortably at two weeks, and now I can nurse her at 3 am without even having to turn the light on... She finds my nipple all on her own and latches properly all by herself!
It's sooo worth it! Your little one will get so many benefits from your milk. You can do it! :)