I just buckle my DD into the seat like an adult. I've been doing that even when she was under 2 and we didn't purchase a seat for her, if the plane was not full and there was an empty seat next to her, I would ask if we could put her in the seat and it was always fine. She is 3 now and gets a little squirmy sometimes but would never dream of unbuckling her belt -- and would not be physically able to even if she wanted to.
This was not really asked in the OP, but I've got a few more tips on flying with a toddler:
If you are bringing a stroller, gate-check it. That way you will still have it to use in both airports. Just take it to the gate (you will have to fold it up to xray it on the belt) and ask to gate-check it, and they will put it on the plane when you board. When you deplane, it will be there waiting for you. So convenient!
One thing that helps us TREMENDOUSLY on flights is a portable DVD player. You can pick them up quite cheap these days - my dad got one for about $50 on sale - and I have to say mine is the very best $80 I've ever spent. It keeps both kids (3 and 8) quiet on the entire flight. I bring a couple of Disney-related DVDs, the planning video, and sometimes will bring along a movie from Blockbuster Online - no due date - so they have something new to watch. DD3 holds the player on her lap for almost the entire flight. Stickers also keep her entertained for quite a while, and small Color Wonder books with a few CW markers or fingerpaints are great because there's no mess. Just remember to only bring as many markers as you are willing to hunt for under the seat when they get dropped!
Many airlines have activity books for kids, all you have to do is ask the flight attendant once you are in the air. Sometimes they also have playing cards too, my DD8 likes to play War and DD3 likes to just sort the cards and read the numbers and count the dots. There are so many easy ways to keep kids busy on a plane without packing the whole playroom, especially if they don't fly all the time and it's a new and exciting experience. DD3 loves to "read" the safety instruction booklet, she calls it "the airplane book" and just loves the cartoon-like pictures. DD8 is fascinated by the SkyMall catalog.
It's really up to you whether or not you want to bring a carseat. I personally would not, unless you are renting a car and will need it anyway. It's just one more big thing to lug around and with a toddler, who needs that?
Have fun!