I just booked our first fight as well! It’s a relatively short flight but I did book a seat for my baby (20 months old). I knew I would freak out knowing I didn’t and that it’s the safest for the baby to be in a car seat in his own seat. I plan on buying the Cosco Scenara Next car seat since that is what I’ve seen recommended being lightweight and FAA approved. I’ll need a car seat anyways once we get there. Besides the safety aspect, I can’t imagine my baby sitting on my lap for an extended amount of time without wanting to get down or squirmy. I’ve gotten some push back from people saying it’s not necessary. It’s the safest though! And I had to put my foot down with my husband since he was under the impression that it’s safe to have him in my lap. He also thought that when we travel to Japan later (from California ), that he would be saving money by having the baby sit on my lap. Uhh no way. Flying free isn’t worth the risk.
We just traveled with our 21 month old and my friends with their 22 month old. We booked a seat for ours, they didn’t, and both did fine! I will say, my son only slept when I held him so if you’re hoping for a plane nap, lap works great. It was nice having the option to put him in his seat when he was awake so he could play and snack, but if your flight is under 3 hours you’ll get through it either way! Over 3 hours I would be nervous about not having the space to let them off your lap.
How long is your flight going to be? I wouldn't stress too much, it will be ok. :) I've taken my little one oversees twice and then flown 3 times within the US. He has not had his own seat for any of these flights. It helps if they take a nap for a portion of the flight. I also brought some new toys and books to keep busy with.