Absolute needed -
Somewhere for baby to sleep which fits in your room, so a cot/cotbed/crib/moses basket/baby hammock/something to prevent them from falling out of bed if you choose to co-sleep.
Something portable for sleeping in/putting the baby down during the day. Either a moses basket or carrycot, reclined bouncy chair or baby swing, or use the pram, or get a really comfy sling. Can also just put them down on a blanket but be careful if on sofas etc because you don't know when they will start to roll (not just off the sofa, but into cushions etc which is a suffocation risk). Leaving them to sleep in the car seat is not a good idea as a rule, though I tended to leave DS sleeping in his if we had driven somewhere. It was the exception rather than the rule though.
Something to transport them around in - car seat essential, whether you drive or not, and a pram/lie flat pushchair/decent sling.
Clothing and a few blankets. Sun hat if due in summer, normal hat (and at least one warm blanket) if due in winter. About 10 complete changes of outfit minimum. One outfit comprises a vest and either a top and trousers set + socks or a sleepsuit (babygrow), add a cardigan if it's cold.
Feeding needs - bottles, bottle brush, steriliser and formula if bottlefeeding. Nursing bras and breast pads if breastfeeding.
Bathing/changing - nappies - cloth or disposable. Cotton wool and a small bowl for water, or washable wipes. Muslins or cotton or terry squares for burp/sick/general spill wiping cloths.
On this front, you don't need baby bath stuff - should be water only for the first few months,
Basic first aid kit - an oral syringe for giving medicine (you get them free with baby nurofen, I think, or can buy them from chemists), baby paracetamol for first jabs, and an easy to use thermometer - the ear ones are good but not that necessary. An armpit one is fine. The forehead ones are useless with a baby because they get annoyed trying to see what you are sticking to their head! Any other medication (infacol etc) can be bought as and when needed.
That is all you NEED. Nice/helpful to have is:
A compatible buggy and carseat for really quick journeys. Also a base for the carseat which stays in the car and means you just click it in and out rather than strapping it in all the time (depends how much you drive I suppose)
A sling even if you plan to use the buggy all the time - means you can hold your baby AND get stuff done with your hands free. Don't just get a baby bjorn/tomy/own brand one - do a bit of research and get a good one. Stretchy wraps or ring slings are my favourite for newborns.
Decent digital camera and a card reader so you can download the photos quickly and make space for more!
Insulated mug for keeping hot drinks hot
Box sets of Friends or other equally long and amusing TV series, for the marthon breastfeeding sessions in the early days.
Tube of Lansinoh if breastfeeding. You can wait and see if you need it, but if you have it on hand you can use it at the first signs of soreness.
Loads of meals in the freezer - things like spag bol and curries and casseroles which can be reheated easily.
Breast pump and a few bottles - useful to have in case you want to leave the baby with anyone. Not essential though. You don't really need a steriliser if you aren't using bottles regularly, you can just wash the bottles really well as breastmilk has antibacterial properties.
Baby swing/bouncy chair type thing. IMO a really basic bouncy chair which is just a piece of fabric stretched over a metal frame is perfectly adequate. They soon work out how to rock it themselves by kicking. I got one from an NCT nearly new sale for £2.50. If you want your baby to sleep in it then get one which is as reclined as you can or which has multiple reclining positions. I am not sure about the ones which vibrate - I read somewhere that there are concerns they may damage babies' joints (though haven't been able to find it again, so I might have imagined it completely!)
Access to mumsnet at all times
Things I found completely useless/pointless:
Bath support - can be useful I am told, but DS always hated the big bath anyway and the baby bath was a pain to use, so he went in with his Dad, I designated this as "his" job since I was breastfeeding, which was the best idea I ever had I think - he got one on one skin to skin bonding time and I got a free half hour to an hour in the evenings
Nappy wrapper/nappy bin. A normal bin (pref. with a lid) lined with a supermarket carrier bag is fine. You don't want to have too many used nappies hanging around, because they stink, even the wee ones.
Snowsuit/padded coat - you can't use them in car seats because it's unsafe (the wadding can compress in an accident and make the straps too slack) and the baby always hates being stuffed in and out of them and they can't move when in one. Also you can't remove it quickly if e.g. going between heated shops and a cold street. Just use blankets until they get big enough to kick them off, and then get a cosytoes thing for the buggy.
Anything is fine to buy second hand except the car seat, cot mattress, and bottles. Some breastpumps are OK second hand - check if all the parts which come into contact with milk can be sterilised or not.