Travel Car Seat
If you're taking a car seat with you, OR if you want to buy one and have it sent to your destination, this is the best option. It's small enough to fit in a plane seat, and light enough to carry through the airport while chasing a toddler. It installs relatively easily (relative to basically any other car seat). And it's cheap enough to leave with grandma and grandpa. It's $58 on Amazon.
I.LOVE. This backpack. It's $125, fits in the overhead compartment, is super light weight, and you can fit so much stuff in it. Like SO much stuff. There's a reason this is the backpack of choice if you're backpacking through Europe for the summer. Or, you know, taking a family of four for two weeks. But I'm a huge fan of backpacks because you won't have to deal with finding elevators, and you can chase a toddler much faster with this instead of wheeling a suitcase.
When your toddler reaches the magical stage of 40 pounds and 40 inches, they qualify for a booster seat. This one is inflatable and you can PUT IT IN YOUR PURSE. We've started using this earlier than the recommended 4 years old because he's hit the height/weight, but know that's not recommended. But we use this for taxis and quick city trips and love it. $26
If you're nursing on the plane next to a stranger, you might want a nursing cover since they can't get away. It's pretty much the only time I use one, otherwise I'd say it's their problem not mine. Anyway, if you want a cover, this one is my favorite. It has a firm material at the top so it bends out so you can see the baby. And it has a thicker trim to use for spitup etc. And the designs are pretty cute. $20 so you don't freak out your seatmate.
$50 for a four pack of these blankets seems absurd. Until you use them constantly, in ways you never imagined. There's a reason there are literally over 1800 reviews and it has a five star rating. We use this to throw over the car seat when/if the baby falls asleep in the car seat. That allows them to connect some sleeps cycles and get you longer stretchs of sleep. You can usually find these on sale at Zuilly, but there's also here on amazon.
The bulk of our toddler entertainment involves him watching my iphone. I've tried a tablet and a kidle kid's fire, and frankly, nothing compares to the ease of an iphone. So he gets mine, although you could repurpose an old one for this too. Either way, you'er going to need good headphones. These are perfect. They're comfortable, have a volume lock so it can't be too loud, and my toddler wears them without objection. So it's a win. $19.