Avent Soothie pacifiers. Many babies will want to suck on something to be soothed, even when they aren't necessarily hungry. These pacifiers have an open back, which lets you stick your finger into the nipple and wiggle it around, convincing a crying baby to suck on the pacifier and quiet down. Touching the pacifier to the roof of the baby's mouth will also encourage the baby to accept the pacifier.
Pacifier clip. We like JJ Cole pacifier clips, which have an easily-attached non-metal clip, which is like a chip-clip.
Naspira Snot Sucker. The concept sounds a bit gross -- use your mouth to suction out your kid's snot -- but they have a filter to prevent mucus from going through the tube, and you're able to control the power of the suction, which makes it quite handy and portable). Before using the snot sucker, it helps to put a few drops of saline into your kid's stuffy nose, so that the boogers become a bit more liquid-y and easy to suck out. Unlike the other popular brand, Nose-Frida, this Naspira has a "real" filter (not just a sponge), and collects the snot in a clear silicone squeeze-bulb, so that you can see how much comes out, then squeeze out the snot into a tissue. The tip also detaches, so it's much easier to wash out. You can sterilize it using a microwave.
Breastfeeding pillow. Whether you plan to breast or bottle feed, you can usually position a few pillows under your arms so that they don't ache too much from supporting baby... but these cushions are specifically engineered to wrap around you so that you can focus on cuddling your baby (for example: My Breast Friends pillow or the Boppy pillow). Once your baby starts doing tummy time, these pillows are also useful to lay under their torsos and encourage them to prop themselves up.
Burp cloths. These cloths are for you to place under your newborn's chin when feeding, in order to avoid getting spit-up on baby's clothes. Place them over your shoulder when burping baby to avoid getting spit-up on your clothes. You can use clean cloth diapers for wiping up after baby. Washcloths can also double as burp cloths. After about three months, you'll probably want to fasten a bib on baby to catch the continuous drool.
Bibs. These really come in handy when baby is around three months old, and the drool really starts flowing. Babies start teething / putting things in their mouths between three and five months old. There are several types of bibs; the ones we found useful had an absorbent front to soak up the liquid, and a plastic backing to prevent the liquid from getting onto the baby's clothes. (For example: these waterproof terry bibs are designed specifically to guard against drool on the shoulders as baby twists and turns.)
Electric bottle warmer or hot water dispenser. Generally, babies aren't too good at regulating their temperatures, so you should feed your newborn formula or breastmilk that's warm or at room temperature. Some parents like to warm up their children's food with a bottle warmer. If you're mixing formula, you can also get a hot water boiler and warmer and mix 1 part hot water to 3 parts cold filtered water (e.g., from a refrigerator).
Thermos for carrying hot water on-the-go. TommyTippy makes a nice one with a detachable container to hold the bottle and hot water.
Fitted crib sheets. Get a waterproof pad like this one for any accidents, and fit a crib sheet over them. Some folks like to put two layers on, in case of a late-night accident, so that you only have to strip off the top layer (instead of stripping off the layer, hunting around for new sheets, putting them on, etc.)
Baby nail-clipper or emery board. Baby fingernails are sharp! When they're just born, you can usually put socks over their hands to prevent them from cutting themselves. After a few weeks, though, you should encourage baby to discover her hands and play with them. You can file nails down with an emery board, or cut after bath time, when they're soft. They grow quite quickly, so you'll probably need to cut them a few times per week. We liked having a baby nail clipper with a light on the end, so that it had a focused light source near those tiny nails.
Aquaphor. Once baby starts drooling/teething, her cheeks may get a rash-like red from the chafing. Wash baby's face with a washcloth+water in the mornings and evenings, then coat her cheeks and any dry spots with Aquaphor, vaseline, or any other lotion that will prevent drying out. Note that before trying any new product on a baby's sensitive skin, you should apply a little on her back first, then wait a few hours or monitor the area for a day to see if there are any adverse reactions. With any rash, call your pediatrician with any questions.
Hooded towels. It's easier to wrap a hooded towel on top of baby's head, then wrap the rest around their body.
Footed pajamas (so you don't have to bother with socks, or two pieces of onesie + pants), which either zip up or have a lot of snaps, so that you don't have to remove the whole thing while changing (for example: this somewhat unisex outfit). Newborn clothing is usually for babies that are 5-8lbs, so a lot of kids outgrow these outfits within a few weeks. We recommend buying clothing for older kids (i.e., 6 months or 9 months -- and "6 months" sizing usually means "for the child to wear until he reaches 6-months-old").
Swaddle blankets (especially important in the newborn days, when swaddling is key); we received these Carter's Swaddle Blankets as a gift, and they're soft, large, and able to stretch over a baby... we also got a Miracle Swaddle Blanket, which worked wonders for getting our baby to sleep in the first few weeks.
Sleep sacks. These are especially useful for newborns, since you don't have to actually worry about their legs -- just zip them up into a sack... for example: these cute zoo animals from Carter's. As an added bonus, these can cover children from 0mos to many months older, because the sack is a fairly flexible fit.
Socks. These are useful not only as socks, but as gloves to keep baby's fingers warm, and to prevent baby from cutting herself with her fingernails). If you do want to use socks as gloves, make sure that the inside of the socks doesn't have a lot of loose strings, since these can tangle up baby's fingers and potentially cut off circulation.
Hats. Keep your baby's head warm. As your baby's head grows, the hats will need to get bigger. (for example, we used this Circo headwear set from 2.5 months to 5 months, which comes with socks that aren't very fuzzy on the inside, and have less loose strings for baby's fingers to get tangled in).
JJ Cole Bundle Me. This is a comfy and warm cover for a stroller. The cover lets you zip up your child inside a fleecy lining -- especially useful for tucking in your baby during the chilly wintry months.
Travel system stroller. These travel systems have an infant carrier that you can put in the car and then click directly into the stroller). It's best to go to an actual store (Babies R Us, Bye Bye Baby, Target, etc.) and try these out to make sure you can maneuver the strollers around. Make sure that both Mom and Dad are able to lift the car seat portion in and out of the stroller -- some of these things are heavy, especially when you add an 8lb baby.
Gate check bag for stroller. Usually at the airport, you can take your stroller through security (on the conveyor belt), but you'll have to check it at the gate. (At your gate, you'll have to ask the customer service desk for a gate-check tag.) To protect your stroller from dirt, you can use a simple bag for your stroller and/or carseat. There are other models that offer more padding, too. Before going to the airport, practice putting your stroller or carseat into the bag, to make sure you'll know how to do it in a hurry, and to make sure your particular stroller/carseat actually fit in the bags. Sometimes, if the plane isn't full, the airline will let you fasten your child in the infant carrier or car seat in the airplane seat.
The Diaper Dekor Disposal System. This really does a great job of containing the stench from smelly diapers... newborns don't stink up the room too much, but, as they get older, they start getting progressively stinkier!
Chux. You can get either small or large. These have a cotton fluff layer and a polyproplene backing, so it's soft on baby's skin, but doesn't let anything leak through (we put these on our changing pad at home, and on surfaces when we need to change baby on the go). Doctors' offices often use these when weighing naked babies, to contain any accidents.
Diaper cream. Desitin and Butt Paste are popular brands (for preventing diaper rash; essentially creates a barrier between the wet poo and the sensitive bum).
Travel changing pad. You can get travel pads from companies like "Skip Hop," which sometimes come as a "changing station" with a travel pocket that can contain wipes and diapers.
Baby bath tub. You'll have to sponge-bathe baby until the umbilical cord falls off, but a lot of kids love water. Target and Wal-Mart sell cheap tubs that fit over kitchen sinks. Another alternative, made of foldable soft firm foam, is the Puj tub. Don't forget to get some hooded towels (to keep baby's head warm when you pull him out of the tub), washcloths, and baby bathwash (this often serves as both shampoo and soap).