I'm trying to think of what we actually used...
Carseat, I've heard not to buy them used either. You never know, and it is safety equipment!
Ring Sling/ moby
swing - we actually went out and bought one, we didn't think we would use it, but it is nice for naps and cooking/ cleaning
diapers/wipes and something to make wipe solution if you use cloth wipes. I wasn't able (lazyness) to use cloth wipes until I picked up a wipe warmer at the thrift store (there were like 5 and they are all .99) Also, flanel sheets to cut up for wipes!
baby powder - we live or die by baby powder :) I use plain cornstarch sometimes
Pack n' play - we co-slept in this (sorta) and it was our changing table, and was really nice for going to grandma's house etc. It folds up pretty small, just the changing table part is bulky to travel with.
I used a stroller to grocery shop. Carrying groceries and baby home was difficult in the summer months, and we didn't have a wagon, and I was told you can't ride a bike with a NB. (Which, I am thinking someone needs to invent a way to put a infant carrier in a bike trailer. I can't be the only person who this affects.)
Clip on high chair. (we don't have the space for those massive stand alone ones)
"Activity center" the little plastic thing with the toys where they can stand and play? We called it a stand-ie. about 7 or 8 mos he just wanted to stand all.the.time it saved my life. DS is a kinda clingy kid, so it was heaven to be able to put him down and him actually enjoy it!
Hand soap - like a 100 gallon drum of it. You wash your hands constantly with a newborn! Maybe some lotion or if you are into it, the disinfecting gel stuff.
I can't think of anything else....