As others have said, an umbrella stroller is a great thing to bring. The bigger strollers are just plain foolish for a variety of reasons, and will bring frustration.
I didn't have a forum like this when my children were young, and on our first cross-country trip with a baby we brought a full-size stroller. It was damaged both by baggage handlers and by me trying to fit it and luggage into the rental car. It took up way too much room in the trunk, but the car seat snapped into it and that's what matters, right?
On our second cross-country trip we used an umbrella stroller. We didn't have to collapse it in the Jetway at the airport or wait for it to be returned and unfold it in the Jetway at the destination airport. It conveniently fit in the overhead bin even with a couple suitcases already there. It took up so little room that we brought it in the car, and that cheap ($10) stroller didn't get damaged at all on that first trip. I did damage it on our second trip, but so what - it was only $10!