Going home outfit (buy one-piece, and smaller than you think you'll need IME!), including hat and socks if it is two piece. Car seat to go home.
While in the hosp, they provide dipes, wipes, clothes, blankets. I took along a NB one piece outfit for her to wear in the hospital but we didn't get around to using it. Couldn't hurt to carry one in case you hate their onesies. Nursing pillow and nursing friendly robe for you and nursing supplies like Soothies and soft nursing pads which not all hospitals have. They usually do have breast pumps if you need one.