The things that I really couldn't do without (I've got 3 kids) were:
cosleeper -- We have the babybunk, but there are less expensive alternatives. None of my kids slept through the night until they were at least 18 months old, and cosleeping is the only way I was able to get a reasonable amount of sleep most of the time. YMMV.
Mei Tai baby carrier: I tried different slings and carriers with all three kids. This one was my favorite and I used it literally all the time with my third.
Car Seat of course... preferably one that can fit into a stroller (if you're getting a new stroller anyway) -- those things get very heavy very quickly. For various reasons, we ended up using several brands. The one with a handle you can grip several different ways was our favorite, because as I said, they get heavy, and it's nice to be able to switch your grip and use slightly different muscles to carry it.
PlayPen/portacrib -- not as necessary at the beginning, but once the baby starts rolling over, it's really nice to have a safe place to put them down while you do something.... like run to the bathroom. It can also substitute for a crib -- my 2yo still naps in his, and we have never bothered to set up the full crib for him. He'll probably be to big for it soon though.
Vibrating Bouncy Seat -- the vibrations are really really soothing to most babies, and this was a lifesaver. Again, it's a matter of being able to put the little one down and keep them happy, or asleep, or whatever, while you do something.
Lots and lots of receiving blankets -- they're great for swaddling, and they're absorbant so their great for wiping up various types of babyooze and protecting your clothes from spit up and such -- much bigger than the little burp clothes and therefore more effective. Also good for absorbing extra breastmilk -- I know that I always leaked from the other side for the first 1-2 months until everything balanced itself out.
The book The Happiest Baby on the Block (or the dvd if that works better for you.) Really -- Watching this dvd during our refresher childbirth class (and I teach childbirth classes) was the most valuable thing in it. Seriously -- watch or read this.
Cloth Diapering -- of all the brands and types we tried, my favorite by far was fuzzibunz. It's a pocket diaper that very rarely leaks, can be as absorbant as you need, and it dries quickly because you take the absorbant "towells" out and they unfold so that they're a flat microfiber towel in the dryer. In my opinion, if you're cloth diapering and doing the laundry anyway, you might as well use cloth wipes too. Their cheaper, more comfy on the baby's bottom, and better for the environment.
Even if you're cloth diapering, do yourself a favor and get a package of disposables to use at the very beginning until the baby stops passing meconium. That stuff is very hard to get out of the cloth diapers.
My kids all loved the baby mat with the dangly toys and later their excersaucer.
I have loved my baby monitor, particularly when the little ones are napping and I'm somewhere else in the house. We still have it on in the kids room. RIght now all three share a room, and if one wakes up in the middle of the night, the monitor lets us get in there and calm them down before it gets noisy enough to wake the others.
That's all I can think of right now. Gotta go get my 2yo up.