If you are transporting a bike via Amtrak, some precautionary notes. Make sure they have a box, in stock, at the station. If not, you will have to find a bike shop and then take a cab to the station. Also, just because there is a station stop for Amtrak, it may not be manned. If it isn't manned, they will not allow you to put the bike, on the train, even if you have it packed, in a box. If they don't have a union baggage handler, you are out of luck.
If I were to get a case for airline travel, I would get one that is almost regulation size. R&E of Seattle has a good case 26x26X10. Some airlines charge a lot for transport of a bicycle. I would break the bike down, S&S couplers or a folding bike. I would also purchase a collapsible trailer, from Bike Revolution, out of Oregon, which makes the trailer for Bike Friday. email@example.com
. The above mentioned together is on the plus side of $600. Unfortunately, the cardboard bike box may be your cheapest option.
One other suggestion. I was able to rent a car from Avis, one way, for a very inexpensive rate, and simply drove myself, and bike to my destination. That solved the problem with what to do with the case.