We used a bassinet for the second but not the first. It was free and allowed us to put off the move to the big bed for a while, but didn't last long as I have huge babies and DD2 outgrew it very fast.

I was a pretty minimalist mum, this was my list of stuff:
Cot and cot sheets (first baby slept with us for first 13 weeks)
64 MCNs, 75 cloth wipes, snappies, covers, liners etc
5 singlets and 4 wondersuits
Car seat
BABY CARRIER - I loved my Close Caboo for the early days, then swapped to an Ergo
8 muslin wraps
Nappy bag with nappy wallet
Breast pads
Bouncy chair/rocker
Pram (not an essential but I really wanted one

I didn't have a steriliser as I never sterilised anything. A few dummies, but we just did them on the stovetop. Didn't need it taking up bench space!

I think that's about it.