(1) If you will often be traveling to places at which you can leave a bike case and your bike route is a loop, then buy a hard-shell bike case.
(2) If you will be traveling less often and/or your bike route is point-to-point, use a disposable cardboard bike box.
If you are traveling on an airline with reasonable bike fees (e.g., Southwest), then fly with the bike. Most airlines do not, however, have reasonable bike fees at this time (maybe some time in the future if we're lucky). So, if you're traveling on most airlines, then use a shipping service like FedEx or UPS. If you are using a hard-shell, you can easily pack and ship it yourself. If you are using a cardboard box, you can either pack and ship it yourself or have a bike shop pack and ship it for you. The latter alternative is about $50 more. I like to pack and ship myself on the outbound leg and have a bike shop pack and ship for the return leg.
Finally, if you only need a bike for a couple of days, rent one at your destination. But check ahead for availability, since places that rent road bikes are not common (mountain and city bike rentals are easy).