This list may not be complete. Feel free to supplement in the comments if there's something important I have forgotten. And maybe there's something I missed that you found life-saving.
Defining "need" may be a bit tricky. For example, plenty of people in the world have babies without owning cribs. Many of our grandparents slept in a dresser drawer. I am assuming a certain western middle-class-ish lifestyle here, that may not be appropriate in all instances. Take it with a grain of salt.
Babies sleep, eat, pee and poop, and play, and they have to be carried from place to place. That is all babies do. That makes organizing this list easy.
1. Crib - Even if you decide to co-sleep, you will want a crib for naps or times when the baby needs quiet time. Cribs should have bars that a kid can't fit her head through. Cribs should not have lead-based paint because many kids chew on their cribs. The latest statistical studies say cribs should not have drop-down sides, because sometimes children are injured by them.
Cribs last FOREVER. Don't buy one new. This is the number one thing I would buy used. I had two kids in cribs at the same time, so a friend gave me an old used one and I bought a fancy new one. The old one has held up better. It is probably twenty years old by now and we still keep it assembled for stuffed animals.
2. Crib Mattress - Well, obviously. Baby mattresses should be firm. Firmer mattresses help reduce the risk of SIDS. Mattresses do NOT last a long time (my toddlers jumped on theirs till the springs sprung). A used mattress also has contagion risks. Buy this one new.
3. Mattress pad - You need two, and they should be waterproof but washable. Babies often pee or poop through their clothes in the night, and you have to strip the bed and make it over again. If you only have one mattress pad, that is a big nuisance. If you don't use a mattress pad, you may have to wash down a poopy mattress in the middle of the night, and that is a miserable task. Sleeping on sheets without a pad can also feel cold for the baby. My kids refused to sleep without a pad in winter.
4. Crib Sheets - Own as many as possible. You really cannot imagine all the ways a child can soil or wet or otherwise make a sheet unsleepable. It's like their super power. Cotton sheets are easiest on a baby's skin. The sheet should fit snugly to the mattress.
My own experience: do NOT buy cotton jersey, that stretchy t-shirt material. I had some lovely cotton jersey sheets and, just like a t-shirt, they stretched out the longer they were on the bed. I heard my daughter screaming one afternoon during nap time, and I ran upstairs to find that she had caught her foot in the loose edge of the jersey sheets, and from rolling around had wrapped it tighter and tighter around her ankle until it cut off the circulation. She ended up being fine, but it could have been a lot worse. Cotton jersey crib sheets are just a terrible idea.
5. Blankets - Babies are not supposed to have loose blankets in their crib, because it is a suffocation hazard. A large, square, flannel receiving blanket for swaddling is perfect. Actually, having MANY large, square, flannel receiving blankets are a good idea. Did you miss that part about pooping through their clothes? That includes blankets. And some babies will only sleep if tightly swaddled, so if you are out of clean swaddling blankets, then welcome to another sleepless night, mom.
A baby under nine pounds cannot maintain its own body temperature, so you may still need other blankets to put in the crib with a baby if the house is cold. They should be large enough to tuck tightly into both sides of the crib so the baby can't pull them loose, and they should not be positioned higher than the baby's armpits to prevent suffocation.
6. Clothing - I know there are a lot of adorable baby clothes in the world. But just because they look cute on the rack does not mean your child will be comfortable in them. Frills and straps and bows and belts do not feel good on baby skin, and the truth is you will be so tired, you won't want the unnecessary laundry. Because did I mention pooping through the clothes? They do that a lot.
What babies need are onesies and sleepers. That's it. Sure, a fancier outfit might be nice for pictures, but truthfully? This is your baby, and you will think he is adorable in anything. Onesies are those little jumpsuits that snap under the crotch. You can buy them in packages of three or six at any store that sells baby stuff. Sleepers are the one-piece long-sleeved jammies. Get footed sleepers to start with, and as your child gets older, he will express a preference for footies vs. socks. I had two kids who REFUSED to sleep in footed sleepers by the time they were ten months old.
And then there is the zipper question. I eventually gave away every sleeper I had that didn't have a zipper. Buttons on baby clothing are a curse and a burden and should be banned by law (okay, not really banned, but I hate them, I hate them, I hate them). That leaves you the choice of snaps or zippers. Zippers are easier, but run the risk of a sleepy or impatient parent zipping baby skin into them. I learned to always zip with my hand between the zipper and the baby's skin. Snaps have no zip risk but OH MY WORD have you ever tried to line up snaps on a sleeper in the dark when you haven't slept in 36 hours? This should be part of the entrance exam for West Point. It tests dexterity like nothing else.
Are you going to breastfeed? I hope so. It's a good idea to at least try it. I breastfed my babies because it was cheaper and easier than the alternative. Here's stuff you need.
1. Breasts - Good news! You don't have to buy these.
2. Pillows - A boppy pillow or some kind of firm donut pillow is a great help. It is not strictly necessary, but when you put it around your waist, you can rest the baby comfortably at breast-level and use both your hands to arrange breast and baby for feeding. This helped me so much that I'm including it on the necessary list. If I hadn't had one, the first six weeks of breastfeeding might have required the help of a second adult. Sometimes it did anyway.
3. Ointment - Breastfeeding hurts. Here's the part where the dogmatists rush in and say "Oh no! Breastfeeding, if done right, does not hurt! You must have had a bad latch." Thank you, dear, for dictating to me how my body is supposed to work and denying the validity of my pain. It's charming of you, really.
My experience: a good latch hurts LESS than a bad latch, but in those first two weeks of breastfeeding, it still hurts. The pain lessens dramatically after about two weeks. I could not agree with Lindsay more on this subject:
If I could choose whether to go through labor and delivery again or the first two weeks of breastfeeding, I’d choose labor and delivery. In fact, I’d choose two weeks of labor and delivery- a baby a day- over breastfeeding. Seriously.
So an ointment to relieve cracked nipples or sooth sensitive skin can help. The ointments safe for nursing (you don't have to wipe off the ointment before your baby nurses) are lanolin-based. You can buy them anywhere baby products are sold.
4. Nursing pads - Your breasts will be bigger than they ever have been before, and you will stuff your bra for the first time since middle school. Welcome to motherhood.
Your milk will not only let down when you feed your baby. It will let down when you hear your baby cry, when you think sweet baby thoughts, or when your breasts are very full. You can keep nursing pads in your bra to soak up the milk so it doesn't soak through your shirt. It's handy. You can buy either washable, reusable pads or packs of disposable ones. Of course, if your breasts are so sore that you are staying inside topless for the first month, this won't be necessary.
5. Nursing bra - The idea is a bra that supports your breasts while having snaps or hooks on the straps so you can uncover your breasts for nursing without taking off the bra. I had four babies and dreamed of a good nursing bra. It is the holy grail of the undergarment world. I suspect it does not really exist. But like the noble mother-knights before you, you must go on the quest. The challenges are many. You can have no idea what size you will wear. Neither band nor cup size is predictable. As your baby grows inside you, it presses your rib cage out, so you cannot try on nursing bras while pregnant and rely on them fitting when you are no longer pregnant. Even if you could, the sizes of the bras themselves are notoriously non-uniform.
After my first baby, I gave up on the whole nursing bra idea. I think the final straw was when I dutifully unsnapped the elasticized straps, fed my daughter and then began to snap the strap again, but the elastic slipped out of my hand and smacked her in the forehead like a rubber band. "No more, " I told myself. "I do not want to tell people some day that my daughter only has one eye because of the Great Nursing-Bra Mishap of 2004."
So I settled for either tank-tops with those little shelf bras in them (which have very little support), or I wore very stretchy normal bras, knowing the process would ruin them. Both were better options than another futile nursing-bra hunt.
6. Burp cloths - you know how often babies poop through their clothes? They spit up on you even more than that. They spit up many, many times a day. And when they get old enough to stop spitting up, they drool from teething. They drool like Saint Bernards. Having a mountain of clean soft thick cotton clothes to catch the goop that flows from their mouths will help you. Even if you reach a point of exhaustion so severe that you no longer care what crusty stains adorn your shirts (or maybe you've given up shirts altogether), your mother-in-law will approve your motherhood preparedness when she sees your mountain of neatly folded burp cloths. This is one of those items you don't even need to buy. Ask your friends and they will give you the mountain of burp cloths they received from their friends.
7. Breast Pump - Not everyone needs these, but if you need one, you REALLY need one, so I'm including it on the necessary list. I tried four or five different breast pumps, and many of them are not made to accommodate the generously proportioned woman. Some of them seemed to be built on the principle that women were creatures with more enthusiasm than intelligence and would buy anything. FWIW, this was my favorite, and I used it almost every day with one of my babies.
8. Bottles - If you use a breast pump or you use formula, you need bottles. And this is an area where the options are so many and so dependent on your baby's preferences, I have no guidance to offer. I would only say since they are expensive, buy one or two and see if your baby will accept it before you spend money on a dozen or on some inclusive set. It may be that the breast pump you like and the bottles your baby likes are from two different companies and don't fit together. Happy pouring, mom!
This list is getting really long with all my opinionated rambling, so I will take a break here. Maybe my commenters can respond to my list so far. I'll cover pooping (they do that A LOT - and THROUGH THEIR CLOTHES), play and transportation later this week.