Hi.

I see you are from Geneva. I actually live just up the road from you and have a 2 yo DD. Who knows...maybe we have crossed paths in real life.
We've traveled several times with our 2 yo DD. With the exception of her first flight, we have always bought a seat for her even when it wasn't required. Most of the airlines will charge you half price for a child under 2. I'm a big believer of car seats on planes and I just think that everyone is more comfortable in their own seat. So that is my first piece of advise. Buy a seat for the baby.
We also still bring a carseat for DD (as recent as a couple of weeks ago). I don't lug our big carseat, but instead, we bought a lighter weight, lower feature carseat (Graco Comfort Sport) that is usually in my mother's car. DD knows when she is in her carseat that she has to stay there. I have had other moms say to me on flights after seeing how well behaved my DD is that they wished that they brought their carseat with for their toddler.
If your child is still on sippy cups, bring a couple of empty sippy cups with for drinks that you can purchase after security. I bring plenty of snacks for DD. The last couple of times we traveled, I brought a DumDum sucker with for takeoff (big treat for DD). When we traveled with her when she was around 10 months old, I remember that a single serve box of Cheerios kept her busy for the longest time.
Another thing that we do is that we try to schedule our flights around nap time, if possible. I think that my DD has managed to sleep for at least part of every flight we've taken. That helps a lot!
As far as activities, don't take anything with a lot of small parts! I usually bring a notebook along with some crayons and stickers, and a few small plushy toys. We also bring a portable DVD player, but have not yet to use that on a flight. We consider the DVD player on the plane for emergency use only!
Good luck with your travels!