Our first couple of flights (our son was about 3.5 months old) we took our big travel system stroller and stuffed it into a bag before gate checking it. We found this to be a big hassle, so we soon switched to using a "snap-n-go" stroller that our car seat fit into. It was absolutely not fancy, but that was nice. The investment was minimal if something were to happen to it during gate check, and it could easily be replaced at our destination if we needed to. It held up very well, though.
Once our son was out of his infant car seat, we switched to one of two options: the Babyjogger City Mini and the Uppababy G-Lite. Both folded easily and were very lightweight. The G-lite is extremely lightweight. We used a very light car seat on the plane, and this could easily hang from the handlebars of the stroller. I think once we put the City Mini into its bag when gate checking. But we never bothered after that. Both the G-Lite and the City Mini have been on countless flights unbagged and have held up great. The City Mini in particular is pretty bulletproof.
Now there are umbrella strollers that can fit in an overhead, so that's an option. Either way, I'm a big fan of going as light and compact as is comfortably possible. You'll have time to figure it out between the two flights. You can see what works and make adjustments as you go.
Enjoy traveling as a family!