Why would anyone rent a bike when you have one of your own? Rentals might require adjustments or learning. If your bike is like a family member, you may want to take it with you.
Aiming to keep things timely by giving some current tips, and receiving yours in the comments, here is a link, courtesy of the IBF, to the International Airline Baggage Regulations for Bicycles showing EVERY current airline and their rates. Many international airlines have no charge for bikes. ALL American based companies have a rate for bicycles as baggage. The cheapest is $50 each way on... actually just use the 'Find' command on the Baggage Regulations and look for $50. Understand that this rate applies to bike container dimensions that total no more than 62 inches. Otherwise extra fees or worse. They'll say it isn't flying with you.
NO regular bike box measures within 62 inches. Only foldable bikes and kid's bikes can pass the test. I caught two mistakes on the list for Frontier Airlines, which charges $75 each way, unless the box is under 62 inches total. AirTran charges $50 + baggage fees each way. Not listed on the IBF site was Spirit Airlines, which charges $75, and Condor that charges $65 per flight. The easy way to find your airline's bike procedures is to locate the baggage info and look for the words 'sporting equipment' or 'bicycles.' Understand that added fees can apply if your item is heavier than average, or longer than 62 inches.
So, the second option is to ship your bike. The old fashioned way was to haul it to the local shop have them undo some of the bike using tools, and reassemble on the other end where delivery to another shop takes place. Pedals stick out as the things you may need specific tools & skill to cope. The going rate for a shop in DC that has no backlog of work is one day and around $60 to $90. The same goes for boxing it up.
If you know a little something about bike assembly, then you could ship it yourself via ShipBikes.com. They send a special bike box called an Air Caddy, triangular in shape with reinforcements inside to keep the cardboard from failing under traveling stresses. The shipping rates from $41 depending on how fast you want your bike delivered. An Air Caddy box is $99. They have a less expensive square box, $35 but it requires total mechanical skill. Either box will have shipping fees to receive it. Here's a review of the service from 2009.
If you are traveling by Amtrak, then a bike box is still required on most departures. Rates are less than a 3D movie ticket. A handful of spaces on couple of trains daily allow for non-boxed bikes (full bikes) at no cost.
In case you didn't know it, there was a time that American travel providers didn't charge, like their foreign counterparts currently do. And there was a time when you could smoke on a plane too. Maybe one of those features needs to come back.