I haven't taken a bike by plane yet, but I know several who have, so the information I'm going to give is based on what those people told me.
First, of the 4 people that I know that have taken a bike on a plane 1 got his bike damaged (which he was able to get repaired but he lost two days waiting for the repair, and another one got his stolen off the belt in Germany! In my mind a 50/50 chance of something happening to a bike is not good odds, even if the two were unusual events there is also the added possibility that the airlines can lose the bike because if airlines can lose luggage, they can also lose bikes.
I know one person who tried to go the rental route, he found out several things. First you need to order the bike rental in advance because in some areas they out of bikes in the tourist season, so you may need to reserve yours as soon as you order your plane tickets. He also said to pack your own pedals, saddle, shoes, clothing, even water bottles and cages as silly as that sounds but where he went they charged him a rental fee for the bottles and the cages! If you'll be bike camping then you need to bring your own panniers or get a rental fee for those as well. Of course you should check with where you're renting from if they charge you for stuff that you shouldn't have to bring along. He got so frustrated from the rental thing he left the bike shop, found another bike shop who sold used bikes, so he ended up buying a used classic bike instead. At the end of his trip he was going to sell the bike but he liked the bike so much he boxed it up and sent back to his home via UPS and he still has that bike.
I've heard on the internet that some travelers send their bikes by some sort of postal service to the motel where they were going to be staying, then called the motel in advance to tell them of the shipment and to hold it in a secure place till they got there (make sure you tip the person in charge of handling your bike at the motel). Once the trip was over they sent it back the same way. No fuss from airlines this way, and you can insure it. What I read on the internet is that people bought a plastic shipping box, had the motel keep the empty box at the motel till they returned from their trip. I also read that some people rented lockers and put the shipping box in a locker then when the trip was over go to the locker and get the box to send the bike back.
Again, I have no experience in taking a bike on a plane, so the info I gave is from those that I knew and what I read on the internet.
Personally from all that I've read and heard is I would probably rent even though I heard of all the charges that they rack up, but I would also consider buying a used bike if a shop was around that did that, or get on Craigslist and buy one that way. But of all the methods of getting a bike somewhere, by passenger plane would be my last choice.