We had soooooo many hand-me-downs for our first that I was finding bags of clothes and accoutrements long after he had outgrown them. We had a very small living space then and tried to pare down to the very essentials but (like you) didn't really know what that meant. So, here is my list, a combination of what we did use and what we will have for the next one (due in Oct)...
Bed- ours, now with a sidecar bed attached for our 2yo DS
clothes- we had SO many, but really he spent most of his time naked or so bundled against the cold that it was all about practicality
diapers- 1-2 doz cloth per size plus a potty for EC
diaper bag- SO would just stick a diaper and a wipe in his pocket, I would throw a couple in whatever type of bag I had with me, and one in the glove box for good measure
wipes- we bought disposable ones to keep in the car, but they always dried out and got dirty (the sticky closure wouldn't stay), cloth worked best for us
cream- my Mama made us salves and powders, but really breast milk, sunshine and air seem to be the best remedies for irritated skin
baby soap- we also didn't use soap until DS was older and actually getting dirty, then it was a drop or two of Dr B's in the bath water
jogging stroller- we just got our first stroller yesterday b/c DS wanted it since he can't keep up on walks and I can only carry him for so long now
carseat- we got one from our friend that was under 5 years old and we knew the history of (no accidents) for the first stage, then got on that converts through the stages to a booster
bottles- someone gave us two, we had them just in case and then passed them onto a friend
stuff for pumping- we also had this just in case and I don't know where it went!
burp cloths- I totally agree with the receiving blankets and prefolds burp cloths
blanket- we got several home made blankets as gifts even without having a shower and we still use them
sling or baby carrier- yes to both, I liked the Moby and still use the Ergo, SO has always preferred the sling
nursing cover- nope
co sleeper- just our bed
boppy pillow- we liked ours, it was also a hand-me-down
high chair- we used the Bumbo with a tray as previously mentioned for awhile, but DS really likes to stand or kneel in "his chair" to eat with us (it is just a chair that I know wont tip and has arms)
swing- DS seriously hated the swing, the bouncy chair, the vibrating thing... all of that
toys- a couple, but he preferred whatever WE were playing with
some type of bathing apparatus- the tub or sink exactly how everyone has described
pack n play- nope
**One thing not on your list that we could not have done without during the first few months was a Yoga ball. There were times when swaddling our DS, holding him close, and bouncing on the Yoga ball was the only thing that would calm him down. My legs were soooo sore, but sooo strong!
I don't know if you have a Freecycle in your area (mine is through yahoo groups) or Craigslist but there is always TONS of baby stuff being given away there. You could get stuff, try it out, and if it isn't for you, send it back out there into the world of Freecycle! Another thing that I always tried to remember was that up until a very short time ago the tiny little being in your arms only had you around it. (S)he didn't need anything except to be held, fed and kept clean. A lot of the "stuff" is for you, and your partner and can be added as the need arises. Nowadays it is super easy to run out to the store, or hop online, and pick up a nursing cover (for example) if you nursed at a park yesterday and felt totally exposed/uncomfortable. Start small and then if you want to get something you will know it is because it will help you, then you wont just be using stuff because you happen to have it.
Good luck and have fun!