Commercial pilot's wife here. Kids and spouse fly free so we fly far more frequently than the average family. And I almost always fly alone with my 4 kids and every trip I learn something new about this crazy aviation transport system.
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I have mixed feelings on car seats on the plane. One of my easiest flights was when the twins were 2 and they each had a seat next to me and nursed the entire time. They kept their seatbelts on and we all just stretched a little. There is no way we could have done that if they'd been in car seats. Also, the seats are 'big' for little bodies. Last week the 4 year old slept lying down on the seat, again seat belt on. A car seat forces your lo to stay in the same position. It also takes up a lot of your assigned plane space. If you don't need it at your destination, I say skip it.
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I don't like the CARES harnesses. On smaller planes in turbulent areas, fine. But on large commercial planes, skip it imo.
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Don't check a car seat or stroller. And if you have to check a stroller, be sure to gate check it. It will be treated slightly more gently!
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I've never run into documentation issues but I do bring birth certificates, adoption paperwork and passports for my kids even on domestic flights. (as a new two mom family with kids I worried a lot when traveling out of the us. I've relaxed a bit in the past 10 years. And again, I've never been asked for any documentation other than passports on international trips)
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Luggage logistics- We don't check any bags. I have each kid over age 2 carry a backpack. It get heavier with age and competence. On my last trip 10 year old ds and I each had a roll-aboard. I pushed one stroller and had 7 year old dd push the other stroller. Strollers can be good (to carry gear or tired kids) but can also be a hassle with escalators, etc.
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My biggest advice-
-acknowledge ALL the staff   (ground crews, agents, the lady cleaning the bathroom) kindly and they'll treat you well.  ("Are you a [city name] based crew?" "This must be the first day of your trip?"- this implies the person looks fresh! "Last leg of the day?" Are all questions that show you know they are more than their job.)
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-Treat it as an adventure. Flights will be canceled or delayed. You may be diverted to an airport you've never hear of. You may exit the plane down a big blow up yellow slide surounded by emergency vehicles. Go with the flow and leave plenty of time for things to go wrong.Â
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-Take the help people offer you. Someone sees you carrying your babe, holding your toddlers hand and pulling a roll-aboard and offers to take the rollaboard on the plane and put it in the overhead bin? Say yes! It will make it easier for you and they'll feel good and invested in your trip. They'll be far more likely to make googly eyes at your kids for entertainment inflight!
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-Buy chocolate at every airport stop!
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Have fun! What an amazing world we live in and to be able to share it with our kids through their eyes can be really refreshing.