Do you have an infant car seat? The bucket-style kind that often go with a travel system.

If that's too bulky, consider getting a stroller frame, like a Snap-n-Go and then you can use your car seat as a stroller. I did this when my first child was still in an infant seat and it worked great at the airport.

Many peds though don't like babies too long in car seats so you want to take a good baby carrier (avoid one that hangs the baby by its crotch like a Bjorn) and alternate between them during your trip.

I assume you're still using the bassinet on that stroller. I can see what you mean about the bulk. I would worry about overheating too in that thing. It looks like a nice stroller for later on though. Can you tip the back enough for a small baby (and leave the bassinet at home?) You could then attach an parasol. Otherwise, bag and bungee cord your Vista and gate check, not luggage check it as the pp recommended. It's not too big for most airlines.