I agree with the bag.
However, my BIL/SIL when they only had one kid, they just jammed the booster (bottom only) into their own duffle which I think they carried on. I don't know what they did when DS came along because they had the car seat/stroller combo thing. But I know it's small enough, I think they just jammed it into their own bag when they got to the airport.
The only thing about checking it is if luggage is lost. When we were in the "bucket" and did lap baby, I checked the car seat for the same reason as PP--too much stuff, especially when DD was tiny and floppy. However, we had a couple times where we had delays and airline nonsense, then I kept it--just in case--and gate checked it once and checked it later once. That last part is two long stories condensed into one sentence, but you get the idea.
Some of that you have to play by ear.
Personally, for how small it is, I'd put it under one of your kid's seats or in the overhead. The bags, while holding other stuff--we put our leftover diapers and toys in it, are huge!!!! So unless you're going to use it for full size car seats, skip it for now.