Take solace that traveling with babies may be easier than you think! These parents also make a good point that traveling with a baby is not that much harder than being at home with a baby. (We found it was actually easier to travel with Adam at 3-4 months than when he was ~1-2 years old, since later he wanted to run around...)
Baby carrier - great for walking around with your baby, as well as on the flight; we liked the ErgoBaby 360 Cool Air; see the gear page for more
Car seat / car seat backpack - see "traveling with carseats" page for details of why we chose to get a car seat backpack
Optional (we didn't bring but some people like):
Hand pump/bottle - gives more feeding options (also some moms like having the pump on long flights, to avoid having to hand-express)
"Lap infant"/"Babe in arms" - Most airlines charge nothing for domestic flights, and ~10% of the ticketed price for international flights, for an infant (less than 2 years old) to sit in your lap.
Other discounts - If you buy your baby a separate seat (which is safer, if you use a carseat), there may be discounts—ask!
Some folks recommend getting a separate seat around ~1 year old (since older babies don't do as well in laps)
If you're using a carseat, get a window seat so it doesn't block other passengers
If your child is 22-44 pounds, consider the CARES (Child Aviation REstraint System) instead of a carseat (>40 pounds they can use the normal seatbelt) --> this is super-convenient and apparently just as safe as a carseat
Ask about bassinets - on long-haul flights, many airlines have seats where you can reserve a bassinet for <2 year old babies, usually behind bulkheads
Seatbelt extenders - we've noticed that many European airlines give out little seatbelts that attach to your seatbelt, and require them during takeoff & landing. There are even some horror stories of airlines requiring parents to take children out of their FAA-approved carseats to use these (which are obviously less safe than the carseats).
Know the rules
Security: the TSA's Traveling with Children page details how security should work (kids don't have to take off shoes and "light jackets", breastmilk/formula isn't subject to the liquids restriction, infants can be carried through the metal detector)
baby food is allowed in your carry-on beyond the normal "liquids and gels" restriction, but you have to separate it out for inspection
In Flight: the FAA's Child Safety page is a bit vague on some points, so check out the better Condé Nast Traveler explainer
Airline Policies: each airline has slightly different rules about how much luggage they'll allow for lap infants, etc...check ahead of time to avoid surprises
Wear your baby - use your baby carrier; most airports will let you wear your baby through security, and it can help your baby sleep on the flight while keeping your hands free; note that most airlines make you take the baby out of the carrier for takeout and landing
you can try bouncing your baby and walking down the aisle if they cry, which can help soothe the baby, and no one row has to hear cries for long...hanging out near the bathrooms/galley can also help
Takeoff/landing - nurse/feed/pacifier to equalize ears
one mom rec's Kali Mur 30cc homeopathic to help inner ears (give just before boarding/takeoff/landing)
travelswithbaby says if your child won't nurse, try a small amount of water in a cup
Toys/entertainment - very young babies don't need much, but older ones might appreciate a new toy
ideas for age-appropriate toys at the Weespring Blog
The Mom Edit has even better ideas...plus some great tips for maintaining your sanity (especially if you're traveling alone with a baby)
older toddlers who don't put everything into their mouths seem to like Wikki Stix and similar clay-like toys
also once they're ~18 months or so, screen time (e.g. downloaded movies on an tablet, or edutainment apps like Endless Reader) is often the lesser of two evils --> just remember to download movies before you leave!
Double diaper trick - one mom recommended putting a reusable cover (like the one you would use for cloth diapers) over a disposable diaper to contain blowouts
also don't forget extra clothes...on one trip, Adam ended up "wearing" a blanket for half the flight!
Sit near the back - sitting in the back of the plane can make for more white noise (even more if it's near the engines), fewer distractions, fewer problems with other passengers, easier to whisk away to the galley, and less stress --> we've generally done back of the plane with Adam and it worked well for us (in most cases we've got at least 1 checked bag anyway, so it's not like we'll be that much more delayed on arrival)
Timing - different babies do better on day flights versus red eyes (if it's a short flight try to time it during their normal nap time), but many say to avoid the "witching hour" from ~4-7 PM
Older babies/toddlers: bring enough food - many flights don't offer food anymore, and even if they do, it's unlikely your baby will like it any more than you do! If your baby is eating solid food, consider bringing a wide array of snacks...we've done squeezable baby food, hard-boiled eggs, crackers, dried fruit, etc.
Benedryl? - Art of Manliness has a good discussion about this
Other Tips
great tips for dealing with security, boarding, etc. on the MD School Mrs blog
traveling with a toddler? Travels with Tykes has advice
Stroller - we have rarely traveled with a stroller, in part due to the convenience of baby carriers and inconvenience of cobblestones/stairs/public transit, but if you're staying somewhere with relatively smooth/flat pavement, small/foldable/light "umbrella strollers" can make sense, and you can easily gate-check them (We got the UPPAbaby G‑Lite, which we use as our normal around-town stroller, and have taken it on a few flights....you can also check out strollers like the gb Pockit, Zoe Travel Stroller, Babyzen YOYO+, UPPAbaby Minu, or Mountain Buggy Nano, which are small enough to fit in most airlines' carry-on)...other ideas at Lucie's List
You might also check with your hotel (if you're staying in one)—some of the nicer ones have strollers you can borrow.
Travel crib - usually co-sleep when traveling, but you might be more comfortable with a separate crib; also check with your hotel to see if they have one you can use; for ideas, see Lucie's List, Baby Gear Lab, Travels With Baby, and Pint Size Pilot (the most recommended seems to be the BabyBjorn Travel Crib Light but Phil & Teds Nest Bed, Lotus Travel Crib, and Peapod are also popular)...for older kids, also consider inflatable "toddler travel beds", described at Travel Crib Reviews
A lot of hotels (and even some Airbnbs) also offer cribs (usually a "Pack-n-Play" style)
Snuggle Me Organic - cosleeper described on the gear page --> when we took Adam on a trip at 4 months he just slept with us in the same bed with no cosleeper, but if we had traveled earlier we probably would have felt better with the cosleeper
Not recommended: My Little Seat - if your baby can't sit up, then just wear him/her or carry them in a carseat at meals...but if they can sit up on their own, this is a highly-rated portable seat that slips on adult seats --> when we took Adam on a trip at 4 months he just stayed in our laps (he wasn't quite ready for the seat); we tried this when he was older but found it wasn't useful...usually we either use the restaurant's high chair (if they have one) or keep him on our laps
Flyebaby - a baby hammock for children under 25 pounds that can be used during the cruise section of the flight, and also doubles as a portable highchair (similar to My Little Seat)...positive review on travelswithbaby.com
note that it has a lot of negative reviews on Amazon, but mostly from people complaining about how it's not a safety device (duh) and can't be used during takeoff and landing (which is true for all baby carriers)
we didn't end up getting this but it's worth considering...
If you haven't already, get a registered copy of the baby's birth certificate
You can apply for a passport without a social security number, but if you have one you have to give it
Even newborns need a passport to travel internationally
This place specializes in baby passport photos: A Foto Video Mail & More, 3041 Mission St (at Cesar Chavez), M-F 9-7 and Sat 10-5, www.notaryphotolivescanshipping.com