Oddly enough, I've found that I "need" more gear with each additional child... for example, with my first child, I could put her down for naps in a little basket on the floor. But once I had a toddler and a newborn, or a newborn a toddler and a preschooler, putting the new babe on the floor wasn't such a great idea!
However, for a newborn in a home without large pets and/or curious toddlers who don't understand the whole "gentle with the baby" thing yet:
--a sling, possibly two different types of sling so that if one has a horrible diaper encounter or catches that spectacular urp you have a clean replacement ready to go. And unless you know from experienc that you love the feel of one baby carrier over another having two different types (a ring sling and a wrap, a pouch and a bjorn-style carrier, etc) can be nice. I thought I'd adore my ring slings but had an unexpected c/s with dd1 and discovered that ring slings "bump" right where my incision was.... but the moby wrap was pure heaven! Until the babe was heavier and suddenly the ring sling was the best. Our local mama store has a "baby carrier rental/trial" program so maybe check to see if any of the shops near you have something similar, or find a NINO group or AP group that has a "sling library" or people willing to let you test drive a carrier.
--diapers! We use a mix of cloth and sposie, and always need more than we expect. (cloth diaper doublers/inserts work great as newborn dipes btw... you can use them later for their intended purpose, but inside a cover they rock the world of tiny tushies). Along with the diapers, some sort of ointment (we like Method brand) or bum spray (we like Kissaluvs Diaper Lotion Potion spray and the older kids like to spritz it on their hair after a shower lol), wipes (we use flannel squares) and a wipe solution (a peri bottle with water and a touch of tea tree oil or grapefruit seed extract works fine, there are also Baby Bits that you can dissolve in a spray bottle or wipe box for use as a wipe solution/bum wash), and a caddy/bucket that can hold enough for 1-2 diaper changes so you don't have to carry a messy babe too far to turn them into a "not messy for the moment" babe.
--puddle pads or something similar. We cosleep (so no need for baby sleep stuff) but I got several yards of that rubber backed flannel from the fabric store and cut them into "toppers" for our bed... this prevents baby poo and breast milk from soaking into the mattress or forcing a change of sheets at 2am. You can buy or make all sorts of waterproof covers for your bed (or the crib)... in general you only need to cover the few feet of bed that the babe (and possibly your breasts) will be threatening. We also keep a few of these pads stashed around the house for diaper changes (we change kiddos on a pad, on the floor).
--car seat. I'd go for the all-in-one style (like the Radian or Britax) that will last more or less forever.
--bouncy seat. They don't have to cost much (anywhere from 20 dollars on up) but they've saved my life so many times. None of my kiddos have liked baby swings, but they've all been happy in a bouncer for the 5-10 minutes it takes me to shower or use the toilet (bouncer right there at mama's feet in the bathroom) or to move the boiling pasta from the stove to the sink strainer (not a good idea to try with babe in sling, even on your back). And they're really small/light weight... we live in a tiny space and larger "infant holders" (swings, pack n plays, playpens, walled floor mats, etc) just don't fit!
--tummy tub. Totally not "necessary", but a friend gave us a tummy tub with our last babe and it was so much easier on my back than anything else I'd tried! I don't know if I would have bought it myself (price ranges from 30-60 dollars so shop around), but it was (is) a fantastic gift item and one I used a lot. We don't have a tub (just a shower stall) and while those flat baby tubs certainly work fine, the tummy tub was a lot easier. It used less water and stayed warm longer, ds loved it, and it was less bending/stretching/reaching for me so an all round win. And some sort of bath product... babies don't need much, but they can get surprisingly stinky if their nooks and crannies aren't cleaned out/aired out. A local herbalist makes a fantastic baby soap with skin soothing herbs, but even something like Aveeno scent free baby wash can be handy for the diaper area.
--a bag for on the go. I only own one bag that was "designed" to be a diaper bag, but I do like switching around the bag I carry every few months so I have a collection of baby/toddler enabling bags... it just needs to be big enough to hold one more diaper than you think you'll need (if you think you need two dipes for the trip, you'll actually need three), the wipes and a wet sack for those dipes, a clean onsie for the babe, a clean t shirt for you, and whatever other necessary items you have (wallet, water bottle, toy/book for older child, snacks for everyone including yourself, etc)
--and then all tha basic stuff like a stack of onsies, a stack of sleepers, a stack of polar fleece pants, a stack of socks (all matching so there's no sorting required), and (since this will be a winter babe, we usually have snow by October) a larger coat for me (so babe can be slung inside with face poking out of my hood) and a bunting for her. Probably a few small blankets to tuck round her in her car seat too (remember car seats are designed for use /without/ coats or covers between the babe and the straps so some sort of blanket or cover is important for cold weather driving).