Hi, J.! The advice already given is perfect.
If you can get your hands on one of those stroller/carseats, I've seen people travel with those and they do seem to be the best option.
If not, we used the carseat on the plane because we figured that our child would be more willing to stay longer in it strapped in because he was used to it from the car. Lugging it around the terminals is a pain and cumbersome, however. We got the $25 carseat backpack thing from Babies R us and it helped greatly. Hubby could strap it on his back and still carry the kid or bags. It also has a handle and can roll if you need. I had to carry it on my back coming back from Maui in the airport once and it was quite a workout!
We have also checked it in while inside this backpack thing and it worked fine. It also leaves lots of room in the seat of it to pack extra stuff.
As for the 3-hour flight - that's not bad, you can talk about everything that's happening at first (flight attendants checking stuff, signs that mean when we have to stay buckled, take off, landing gear going up, etc.). Then you can read books or watch a DVD if you have a player. Snacks help. It's always fun to have a NEW book or a toy to add excitement or distraction. The Leapster is a great (much smaller) item that can read stories to her, play videos or she can play educational games on it when she's a bit older.