Depending on your mode of transport. For all private hires (Grab, ryde etc), all these requires baby car seats. For taxi or if your hub/friend/family is driving, can just carry baby on hand. The nurses should be able to guide you on how to carry or hold your baby during your stay :)
I cannot advise on the hospital bag part as i gave birth in TMC. I think you can find their list online.
https://www.kkh.com.sg/patient-care/your-hospital-stay
https://www.healthxchange.sg/women/pregnancy/hospital-bag-for-delivery-what-to-pack