Every hospital provides different things for mom and baby during your stay. My hospital provided diapers, wipes, asperater, rectal thermometer, and large receiving blankets for baby. Everything came hone with us except the blankets (we kept two). For mom, they provided mesh underwear and pads post-delivery, a gown, all shower supplies (shampoo, conditioner, and soap), plenty of towels/wash cloths, a pericare bottle for cleansing the downstairs after using the bathroom, and tooth brush and paste if i had needed it. Also, most hospitals provide formula if your baby can't/won't nurse or if you don't plan to nurse.
I would call or visit (maybe take a tour) your birthing center to find out what they provide. My husband and I WAY over-packed for our first baby. We will lighten the load this time around.
As far as essentials needed after you being the baby home, I would plan to purchase diapers (size NB and size 1 at least), wipes, clothes (not just newborn size as your baby may be too big or too small for that size), mittens to keep baby from scratching his or her face, socks or footed pants/pajamas, a nursing pillow if you plan to nurse (not essential, but very helpful), nipple cream (your breast friend if you do nurse), breast pads because you're going to leak (even if you don't nurse), swaddlers or thin blankets to swaddle baby in those first weeks, one or two pacifiers (dont buy a bunch.. baby may not take one brand and may live another brand -- many babies like Avent brand Soothies), bottles if you plan to bottle feed breastmilk or formula (my favorites were Avent brand), somewhere safe for baby to sleep such as a crib or pack'n play, and maybe formula if you're not planning to nurse (another item that you don't necessarily want to stock up on just yet.. baby's tummy may not agree with certain types and that is too expensive to waste).
That's all I can think of at the moment. Lol I hope it helps some! Good luck!