Can I try out a flight booking API before I start selling with it?

Yes, in nearly every case you can. Most flight booking APIs come with a test environment, sometimes called a sandbox, where you can run searches and make bookings that are not real. Nothing is ticketed and no money moves, so you and your staff can practise without any risk to a customer.

What you see in a test environment is not always the same as what you will see when you go live. Some suppliers return made up flights on a fixed set of routes, so the timings and fares can look strange and you should not read anything into them. Others give you genuine search results and only block the final ticketing step. Which of the two you get depends on the supplier and on the agreement you have with them, so it is worth asking plainly rather than assuming.

The purpose of the test stage is to prove that your side works from start to finish. A search should bring back something you can put in front of a customer, a booking should come back with a reference you can find again, a cancellation should behave the way you expect, and the people who will use the screens every day should have tried them. When you are happy, going live is usually a matter of swapping the credentials rather than building anything again, although some suppliers ask to look over your site first.

Ask your supplier whether test access is included in your plan and whether their test data is real or simulated, then run your most common routes through it a few times before you let a paying customer near it.