We didn't use a specific nappy bin either, we just have a bog standard bin with a tight lid that the nappies go in.
I'd say the only "essentials" are -
a good supply of basic clothes (supermarkets do great multipacks of vests and sleepsuits etc).
Somewhere for the baby to sleep - whether that's a cot or co-sleeper etc. They grow out of moses baskets fairly quickly, we never had one but DID have a really expensive sleepyhead!
A changing mat (we never had a changing table, just did it on the floor, but it depends how your back is!).
Muslins for puke/spit/pee incidents.
Nappies/wipes/bum cream.
A carrier or buggy - we didn't use the buggy at first, he just went in the wrap and I carried him because he was a clingy baby.
A car seat if you drive.
Obviously you end up buying loads more, and lots of things make your life easier, but those are the "essentials".