I just did this last week - flying to FL with my 4 year old & 2 year old. I actually put his carseat on the stroller. It seemed pretty secure, but just to be safe, I bought some bungee cords (at Target) and used them to actually strap the carseat to the stroller. It worked perfectly and getting through the airport/security was much easier than I thought. Also the bungees were easy to strap on & take off, and were small enough to store in my carry-on bag.
There is also a product that I saw on Amazon called "Traveling Toddler Car Seat Travel Accessory" for $14.99, where you can strap the carseat on to a carry-on piece of luggage. Then you can just check your stroller (most airlines its free). This also seemed like a good option, but I tested out strapping my carseat to my stroller at home and it worked great, so that's the option I went with. I also liked having the option of the basket under my stroller, in case one of the kids got tired pulling their backpack (advice I got from a Mom on this site).
The whole idea of getting through the airport & through security, while lugging our stuff - and - keeping an eye on both kids, was really stressing me out pre-travel. But I practiced at home (also had my kids practice wheeling their little carry-on backpacks), gave myself plenty of time at the airport, took a deep breath, and was very patient. It really went off just fine and I found people were very helpful & friendly - most people said they'd "been there / done that"!!
Good luck with the travels - and have FUN on your trip!!! : )