I ride car-free trails almost exclusively. I am attempting to ride in all fifty states. In order to accomplish this I sometimes travel by air to get to the trails I would like to ride. Here are some of the things I have learned in the process.
The first choice to be made is to transport my bike or rent a bike at the destination. I have done both and each has both advantages and disadvantages. Let's consider renting. Not many bike shops rent bikes. You are most likely to find a shop that rents in locations that get a lot of tourist traffic. Even if you are able to find a rental then you need to ask what type of bike does the shop rent. Often these shops rent beach cruiser type of bikes not suited to longer distances most regular riders prefer. I recently rented a bike while visiting Florida. The shop had a choice between beach cruisers and road bikes. I normally ride a flat bar hybrid. I figured since the trails I planned on riding were paved, the road bike should be fine. It was fine but I really missed my own bike by the end of the week. Some people mitigate this by bringing their own saddle, pedals and helmet.
I don't have any issues with riding a different saddle for a few days but some people are very particular and really want their own. Since I use flat pedals this is not a problem for me but if you clip in then you may want your own pedals. I bring my own helmet since I have by Gopro mount affixed semi-permanently.
Logistics are another issue. Airline arrival time plus baggage pick up, rental car pickup and travel to bike shop may result in arriving after they close. You may require an extra day on both ends to allow for this. Speaking of cars, I don't know of any agencies that have cars with bike racks. You will need a vehicle large enough to put your bike inside. Remember this when your are reserving a car.
Sometimes it is not possible to rent a bike in the city you are traveling to so you decide to take your own. I have used and suggest a hard case. Packing your bike is not trivial. You need to remove both wheels, the pedals, the handlebars and the rear derailer. You can let the grips, shifter and brake controls attached to the handlebars. You just need to rotate the bars and position them to make everything mostly flat. Attach spacers to both wheel attachment points on the frame to prevent damage. Local bike shops will have these from new bikes that arrive at the shop. They are normally discarded. The wheels are placed between layers of foam. The frame and remaining components are then placed on top of that with another layer of foam on top. Then it is just a matter of squeezing everything tight and strapping the case closed. If you do this yourself remember you will need tools at the other end to reassemble. Don't forget the pedal wrench. When bringing your bike as a luggage, remember that you should not pack CO2 cartrages. These are not permitted per TSA. Also your case may be opened for inspection by TSA.
Airlines often consider bikes sports equipment and only charge as it it were an extra bag. This is great but will likely have a weight limitation. When I did this the limitation was 50 pounds. That may sound like a lot but it includes the weight of the case. I was overweight with my carbon frame bike if I included the saddle and tools. My aluminum frame bike wouldn't fit the case.
If this seems like too much trouble you can take your bike to a local bike shop for disassembly and packing. Then you can ship it to a local shop at your destination for assembly. With this option you are back dealing with operating hours of the destination shop.
Now let's consider the fun part, riding. When choosing places to ride remember you will need to return a rental bike to the place you got it from. If it is your own bike, you can depart from a different location than the start point. This will require you to transport the hard case, incur additional charges for one way car rental and higher ticket prices for your airline ticket. Even if you return to the same city, consider what you will do with the case. Will the hotel or bike shop store it?