To the hospital bring birth announcements, shampoo, body wash, comb, brush, deodorant, make up (for some people) lib balm, slippers, pillow, coming home outfit, book, nursing bra and pads, slippers or flip flops, old underwear (i bleed a lot and do not want to worry about it) If you bring your own gown make sure it buttons all the way down for breast feeding.
For baby a coming home outfit, camera, receiving blankets, t'shirt, booties, optional: hat, mittens, pacifier, car seat, (best to install before the birth to take your time for proper installment not in front of the hospital) sometimes a car window shade, ID or DL or ask for a bracelet matching the baby.
Dad should pack a bag too.
For him book, phone,, change, toothpaste and toothbrush, change of clothes, comb, brush, ID or DL or ask for a bracelet matching the baby.
I have 2x gone to the hospital because my water broke and once there they will not release me until labor begins or they induce at 24 hours bring things that entertain you like books, music, lap top, knitting, etc for mom and dad for a while. Also my last 4 babies had to go under the lights for jaundice. I was allowed to stay in the hospital with them but it was very boring and lonely for three additional days. If I left I had to leave the baby so I choose to stay. My hospital would not allow me to leave the floor.
I also demand (yes demand) that I not be put in a room with a teenager or a c section. The teens sleep with the TV blasting all night and tend to bring everyone in the HS to see the baby in the hospital sometimes with foul mouths. The rooms are small and the curtain does not give enough privacy and silence. C-sections have to be monitored by the nurses and often this means waking the patient several times during the night. (I will have no help when I get home so I like to rest as much as possible at the hospital.
It is a good time to speak to the pediatrician about circumcision or ear piercing if you plan to do it. If you need referals for one or info on birth control they can give that to you as well. If you already have a pediatrician make sure the hospital notifies him of the birth or you do, so he/she can make the first visit to your baby at the hospital.