Carseat (if driving) or stroller or carrier (for check ups).
Cot - Optional cause I bought mine and she has only slept in it for a day…. It’s way easier for us when she cosleeps. (Will put her in cot soon when she starts flipping).
Bath or daily lotion - up to your comfort level, some mummies don’t shower their babies, only wipe downs till around 2 weeks / 1 month.
Bath towel & burp towels - Can buy in bulks, it will be useful.
Wet wipes - If you are ok, then prepare more wet wipes. Or if you intend to use cotton wipes with boiled warm water, then invest in cotton balls/plain wipes :)
Diapers - Try out samples that you have on hand to see if baby is alright with it then purchase in bulks. Hospital usually uses Huggies JB/NB.
In conclusion… there’s no what to buy first but rather what you intend to let baby use first. 😃