As a disclaimer, I'd like to say that purchasing a seat is the safest option for anyone of any age. However, most airlines do not require you to purchase a seat for a child under two years of age. They are considered a "lap child" and do not need a ticket but must be seated on a parent's lap. There is usually no charge for a lap child but if you're flying internationally, they will usually charge you a nominal fee to bring a lap child.
Even if you do not purchase a ticket, I would recommend calling the airline to notify them that you're bringing a child under two so they can indicate on your ticket that you will be traveling with an "infant in arms." You can do this after you purchase your own tickets online. This will help when going through security. Another handy thing to bring to the airport is a copy of your baby's birth certificate (passport is a must if traveling internationally). Not all airports or airlines require a copy of the birth certificate to verify proof of the baby's age. However, to spare yourself the agony of figuring out which airports or airlines require it, I recommend just bringing it. You don't want to be caught at an airport (like we were) that required it and possibly not making your flight because of it. Our baby was only 3 months old yet they needed "proof" that he was under two! It caught us by surprise since he had been on another flight at another airport and they did not ask for any "proof" of age.
If you need an infant car seat at your destination, I highly recommend bringing your own car seat rather than renting one at your destination. Rental car seats can be dirty and you have no way of knowing if it had been in an accident. Most airlines will usually let you check in the car seat at no extra charge with your bags. However, since airlines fees and rules are constantly changing, I'd recommend calling ahead to make sure. When you check in the seat, they will usually put it in a plastic bag. I've also only checked in the car seat base without charge when I wanted to use the car seat at the airport.
Since I'm traveling with a infant car seat, I did not bring a full-sized stroller. I just used Baby Trend's Snap-N-Go. I liked this model a little more than the Graco because of the compartment that can hold keys or miscellaneous items. This was a lot easier to use than a full-sized stroller. We've even rolled this on sand at the beach in Hawai`i. You can gate check both the car seat and stroller at the gate. However, if you're gate checking your car seat and stroller at the gate, I would recommend getting a stroller bag. I used to just let them put it on the plane as is. Unfortunately, during one of our flights, it was raining when we landed, and the stroller and car seat were soaking wet by the time they brought it up to the gate. Since I needed the car seat for the car ride home from the airport, my baby had to sit on a wet seat.
I also prefer to gate check the infant car seat in the event there is an extra seat on the plane. If there is an extra seat on the plane (I usually check before boarding), you can use the car seat to place your baby. Just make sure there is a label on the seat indicating that it can be used on aircrafts. It should say, "This restraint is certified for use in motor vehicles and aircrafts." They will only allow you to place a car seat on the window seat (or middle seat if you are in a row without a window seat). The car seat cannot obstruct a passenger from safely leaving the aircraft in an emergency. Of the 30 flights that we have been on, there have only been 5 instances where there was an extra seat available so it is not a common occurrence.
I would also like to remind you that most airlines will allow you to do early boarding when traveling with a child under 2. TAKE ADVANTAGE OF THIS EXTRA TIME. It will give you a chance to put your stroller and car seat in a gate check bag and give you time to situate yourself on the plane before the "rush" of other passengers. You can also ensure you will get that coveted overhead bin space since you're now carrying more stuff for the baby. However, I would recommend using anything that you may think you'll use frequently during the flight to be stored under the seat in front of you. It is a hassle to have to constantly open and close that overhead bin when in flight.
I've always been a fan of baby wearing. Baby wearing is especially helpful when traveling. However, if you're going to attempt baby wearing, I would recommend using it with your baby for at least several weeks or days beforehand so they're accustomed to being in one. For the first four months, I used the Moby wrap. I wore it everyday like a shirt and started when my baby was a week old. I used this while flying as well. TSA will not make you take it off and you can request a "pat down" screening to avoid having to take it off. Once the baby was four months old, we switched to the Ergo. It is awesome in that both parents can wear it. I've breastfed with both wraps on. Both carriers are great when hanging out at the airport and when walking down narrow airplane aisles. Helpful to use when walking around during the flight as well. The only drawback is that they will not let you use it during take offs and landings. I never really thought much about it until I saw this video which came from the NTSB website...
What you bring on the flight depends on if you're breastfeeding or bottle feeding. I breastfed for 22 months so we did not have to bring any bottles on our flights. It was awesome. I could feed the baby (under a nursing cover to prevent making anyone feel uncomfortable) on demand and there were no bottles to clean or formula to warm up. Breastfeeding during take offs and landings were also helpful to prevent any pain that may be caused by the change in cabin pressure. It was also very soothing for the baby and helped with a change in surroundings and time zones. If you're planning on having someone bottle feed your baby while you're at your destination, you may want to pack a bottle and breastpump in your check-in luggage. It is usually not necessary to bring with you as a carry-on.
Since I have limited experience with a baby on formula, I can't say much other than to bring bottles, more formula than you think you would need in case there are any delays. To prevent any ear pain from changes in cabin pressure, you can use a pacifier or feed the baby a bottle.
If your baby has transitioned to solid foods, I would recommend bringing jarred baby food. Most TSA security checkpoints will let you bring jarred baby food as long as you declare that you have it as you're going through your screen. They may take it to do extra screening. I'm a big fan of making my own food but when traveling, it is easier to bring the jarred stuff since you don't have to worry about refrigeration or food becoming spoiled.
As for what you should pack in your bag other than food, I would recommend bringing at least three changes of clothing (don't forget the bibs). You never know how many "pooplosions" you're going to get and need to change the baby's entire outfit. If space permits, it's also not a bad idea to pack an extra set of clothing for yourself in case of an accident leaking onto yourself. I forgot to pack extra pants once and my baby had to spend the rest of the flight without pants. I'd also pack an extra large ziplock bag to store the soiled clothing to minimize smell and soiling the other things in your bag.
I would also bring any comfort items like toys and pacifiers. I like to tether my things so they don't fall on the nasty airplane floors. I loved a product called "Secure-a-toy" for everything from my baby's wubbanub pacifiers to Sophie the Giraffe. Those were honestly the only two "toys" that we needed in the first months of flying. Something as simple as an empty beverage cup provided by the flight attendants can be a great "toy."
I also packed a blanket. Depending on the time of the year or the destination dictated which blanket I brought. When traveling to warmer climates, I brought a Aiden and Anais Swaddle blanket or a ROSK tieable blanket. Both of these blankets are awesome because they can double as nursing covers or covers for your infant car seat.
I also packed more diapers and wipes than I thought I would need on a typical day (traveling can sometimes give a baby diarrhea or constipation) and you don't want to be caught without diapers. I also use a changing mat and made sure that was stocked as well.
I would pack the following items separately in its own ziplock bag for ease when going through security... I packed Lansinoh if breastfeeding (even if I haven't used it in months) since my baby tended to nurse more than usual when we traveled. I also packed (and this may be a bit OCD) a tiny squirt bottle of rubbing alcohol. I just bought an empty 3 ounce squirt bottle and filled it up with rubbing alcohol. I find airplanes to be absolutely filthy and I would take the preboarding time to clean the arm rests, seat back table, and window sill and shade with the alcohol. Rubbing alcohol has a distinctive smell so using early boarding to do that will prevent any weird looks. LOL! I also brought a travel-sized bottle of alcohol based hand sanitizer. As a parent, you can never have too much of that around.
When your baby is an infant, it is probably the best and easiest time to travel (until they're in college). LOL! Since most are not sleeping through the night, they adjust easier to time zone changes, don't require food other than breastmilk or formula and you can use a carrier to bring them anywhere a stroller and a heavier toddler may not be able to go to.