Here's what I find useful:
car seat - I like a baby car seat for the first few months, then a reversible once they grow out of the baby one. Dinosauria used the reversible one in the rear-facing position for a few months, then in the forward facing position until she outgrew it.
stroller - Am I crazy to have three strollers? I think probably not really. Sort of. There is one bare bones frame that has a basket underneath and the car seat clips on to bars. It's lightweight, very easy to get in/out the car, very manueverable. I can open/close with one hand. That works brilliantly in the first months. I've loaned it out, gotten it back, it's on it's 3rd kid and showing no signs of breaking down. The lightweight stroller is bigger but works well once they're out of the baby car seat. Our other stroller I call the 4WD stroller. It's a big-ass Phil and Ted's and it powers through leaves, mud and snow like you wouldn't believe. It has been hanging out in the garage for the last couple of years, but it is going out tomorrow. Tuesday the frame-stroller got stuck in the snow on the way to/from Dinosauria's school. We did have an umbrella stroller - its wheels liked to go in 4 different directions at once, making it hard to steer.
high chair - great social tool :) Baby Beluga won't be offered solid foods until later this month, but she loves to sit at the table and "talk" with us. She'll also snooze in it, especially when she's got a cold and needs to have her head up a bit. Ours reclines. With her in that, I can use both hands to eat rather than holding her and trying to eat with one hand.
baby carrier - you've already heard why I love this.
baby gym and exersaucer: the baby gym I bought on the advice of Dinosauria's pediatrician, and found it really helpful because she would happily play under it while I took a shower. The exersaucer I bought at a sale and it worked great as a "keep her happy in one place" toy.
Sleeping: Some people do co-sleeping - we don't because I'm a heavy sleeper, medication can make me sleep even sounder. Baby Beluga sleeps in a playpen next to me but will eventually move to her crib (currently in the basement). I have 3-4 crib sheets. No blankets in the crib/playpen, but I have a huge box of them downstairs. Dinosauria has her favorites in her room for making forts; Baby Beluga will get some of the box for fort-making when she's a bit older and the rest will go to charity. We have a nap nanny that I bought when BB was a newborn with some of the gift cards we had been given. She was having an awful time with reflux, a cold, and a reaction to her bug-infested formula. The formula was recalled and most of her tummy issues were resolved within 72 hours of being on liquid formula. I still use this when she's got a stuffy nose.
We have a little machine that plays recordings of noises - waves, rain, hearbeat, nature sounds/whales. BB loves this.
Diapering: a diaper bag or pail, depending on whether you use cloth or disposables. We use both, so have both. We use disposables at night. During the day, cloth with water-proof covers. No pins - most either have snaps or velcro, but there are also snappis that we use. A pad to change the baby on - although you can just use a folded towel. Some wipes - either disposable or wash cloths that you throw in the wash. A small backpack to use on outings. You don't need a huge diaper bag. I'd also suggest skipping cutesy ones if you want your husband to carry it for you :) I use a small backpack. I stock it with a pad, a couple of diapers, wipes, a change of clothes for Baby Beluga and bottles. I have hand sanitizer in the outer pocket. We have a tube of diaper ointment we use as needed, but not at every diaper change. If you're interested in cloth diapering, let me know and I'll post links to the types we use.
Feeding: I combifeed. I was advised to buy a hospital grade electric double pump to increase my supply. Since my breast reduction was so extensive, this didn't work. My supply was what it was, is what it is, and no amount of pumping will increase it. We had too many bottles. We've got it down to 16, I think. That gives me bottle washing every other day if I don't get it done each night. Most of those are 260ml, a few are 125ml. We gave most of the 125 ml to our local Social Services dept when Baby Beluga started taking more than 125 ml at a feeding. She nurses, then takes her bottle, then sometimes goes back to the breast afterward. The second time at the breast is more for sucking than actual feeding.
I have about a dozen flannel prefold cloth diapers I use as spit up cloths. They are soft, she can chew on them, they are super absorbent for when she spits up. I put one under her head at night in the playpen. For newborns, they make great little covers when they are in the car seat and you just want to keep the sun off them or it's not cold enough to put on something really warm but you want to tuck something around their legs to keep them a little warmer.
Later on when they start wanting to feed themselves, some plastic bowls come in handy. Bibs are good for feeding and drooling.
For newborn: 5-7 snap-side tshirts, jammies, diapers, a few little outfits, socks. Depending on the weather where you live, you may not need the outfits or might need a snowsuit. The snap side tshirts are because they don't press against the umbilical stump while it's healing. Some newborns are really big and might not fit in newborn size. Some are tiny and need preemie. Mine needed preemie, then newborn, then graduated to the bigger sizes. Buy sparingly in the smallest sizes - you can always get more stuff if you need it.
We had three newborn size "swaddlers" that I'll be happy to pass on, if you want them. Two are cotton, one is microfleece. I promise none are pink. They are these:
Post-newborn through toddler, e.g. after they've outgrown the newborn size stuff but before they head off to school: 5-7 shirts, some pants/shorts, socks, 2-3 pairs of pajamas, a jacket, a sweater. Shirts that snap between the legs are great because they don't ride up during those early years. They don't need shoes until after they start walking. I love overalls for little walkers and crawlers.
Bath gear: we have a baby bath tub, but it's not really a necessity. You can wash baby in the kitchen sink if you want. A couple of towels - you can use what you have rather than buying hooded towels and special baby wash cloths. The money we could have saved there... so sad. I have very few of the stuff you find in the giant aisle of baby lotions and potions. In our bathroom are a small bottle of baby shampoo and body wash, a tiny bottle of baby oil, and a tube of baby lotion. I shampoo BB's hair once a week. When she has cradle cap, I use a soft-bristle tooth brush and some baby oil to loosen the flakes from her scalp before washing it. The lotion I use if I think she's getting dry skin.
Baby toys - I find most of them were never played with. Really. A few favorites, everything else can go. We've got way too many of them.
Baby first aid kit - a nasal aspirator (aka the Snot Sucker), a bottle of saline drops, a rectal thermometer, something to bring the fever down (we use Baby Tylenol drops) and a humidifier. You want to have these before baby gets sick the first time so you aren't out at 10PM trying to find them.
Books and music: can't get enough books. Some kid music is ok, other stuff drives me batty. My rule is they have th listen to some of my stuff if I have to listen to some of theirs.