What do I absolutely need to get?

First I would say stay away from babiesrus if money is tight. They tend to be over priced and you can find most of everything you need at Target or Walmart much cheaper. What do you have so far and when are you due? Have you had a shower yet?

When our daughter was first born,all we really ended up using was the boppy,carseat,of course diapers and wipes,and clothes.Everything else really wasn't needed.We breastfed and bedshared/co-slept.By the absolute essentials,and then add as needed as the baby grows.

Blankets
Clothes
Diapers- cloth or disposable
Wipes
Your boobs
Pump and bottles if you want. Or formula and bottles
Bed unless you plan to cosleep

Wow! Super helpful already!

Never ever enough diapers and get different sizes cause they grow fast.
Bouncy seats with vibrate option are a miracle for fussy babies.
Burp clothes
Blankets
Warm clothes since weather is turning cold.
Look for a stroller skeleton, all it is is metal bars made by graco that ur car seat can click into all the other ones are way bulky and take up ur whole freakin trunk and are a pain to deal with. This one folds flat and is super easy.
I suggest a travel high chair instead of the tall ones. Travel high chairs are cheaper and take up less space and they strap to a normal chair, if I could go back I'd never get that big bulky high chair.
Diaper genie 2- it's a god send!
A baby bathtub unless u plan on bathing with them.

I would get some used stuff if money is tight. Just get what you think you need for the first few weeks, you can add on later. Having a baby doesn't mean you are stuck inside for years, LOL. You won't need a high chair right away, or a jumparoo, or a crib if you have a bassinet (usually last about 4 months). You won't need bowls or spoons, or anything for feeding till about 5 or 6 months. Diaper genies are money pits, those refills are expensive and you can just put diapers in plastic grocery bags and take out the garbage at the end of the day. 
You don't need special baby detergent, use your own, baby will be next to your clothes anyway. You will see any sensitivites in the first week and then you can adjust from there.
Get your car seat if you don't have one already, buy NEW never buy a used car seat unless you know the exact history.
Don't buy a big giant bunting coat thing, they can't be used in car seats and are a waste of space and money. Get thin fleece jackets, sleepers, etc, and layer. Blankets are a must in the car anyway.
Also don't buy a car seat cover for the winter unless it is the kind that goes over the top of the seat, like a shower cap. They are like $15 at walmart, vs $40 for the dangerous bundle me ones at babies r us.
Good luck and congrats!! 

 How far along are you? Don't go to BRU - they're overpriced for things you can get at Walmart, Kmart, Target, etc., for things that are just as good in quality and just as cute. What do you have right now? Anything?

We are on our sixth baby. We have a carseat, a crib highchair, clothes and blankets. In the past we have had the other things like swings and bouncy seats, but they take up a lot of space and many babies dont use those items for long.
Try craigs list and mom to mom sales too.

For low costs? I'd go with your boobs, secondhand clothes (Kijiji often sells in bulk. $20 for a garbage bag or something like that) cloth diapers (You can get amazingly cheap prefolds and covers- around $1 each for the prefolds) and somewhere to sleep. A crib or floor bed is a good option.

If EBF, it will be that much easier for you. Not many supplies and you don't ncessarily need a boppy, either.

Carseat, diapers, wipes, bassinet, onesies/sleepers, socks, booger sucker (hosp will probably give), blankets, baby nail clippers (I used mine at least once a week), baby bathtub and soap.  If baby gets colic, you'll want to have one of those vibrating bouncers or one of those large exercise balls.  I'd hold my LO and bounce on that and it helped calm her.  BRU is very pricey.  Definitely check Walmart & Target.  Many of their items are online only stuff.