There are almost no absolutely free APIs in the travel industry. One way or another, you have to pay for product use. That said, you may enroll in affiliate programs to share commission fees with a provider or just try sandbox versions for your MVP. Here are some popular APIs that you can get kinda free or with shared fees.

Travel Innovation Sandbox by Amadeus
Sandbox APIs include: flights, hotels, cars, trains, and points of interest.
How? Just register, get your API key, and you’re good to go.
Limitations: No commercial use. No more than 5 concurrent transactions and no more than 5,000 calls per user per month.
Use cases: PoC, testing, student works.

API Explorer by Sabre
Flights, hotels, ground transportation, and cruises. Additionally, fare ranges, low fare forecasts and history, top flight destinations, and air travel seasonality.
How? After you fill out the test API form, the Sabre account manager will contact you and provide you with a token.
Limitations. No commercial use. You get access to REST APIs only, while Sabre also suggests a set of SOAP APIs. Depending on your profile, some APIs may not be available at all.
Use cases: PoC, testing.

Skyscanner API
The API includes info on flights. As soon as you generate over about $1300 worth of revenue per month, you can start earning.
How? You directly request API access from Skyscanner.
Limitations. Only info sourcing with no booking. They’ll vet you. Live flights calls are limited to 100 per minute, flights cache up to 500 per minute, car hire up to 100 per minute.
Use cases: Metasearch startups.

OpenSky API
You get real-time flight and aircraft info.
How? It’s open and can be used right away.
Limitations. Noncommercial use.
Use cases. PoC, student works, nonprofit projects.

Booking.com API
The API is mostly aimed at accommodations. You become their affiliate partner and receive part of the booking fee.
How? Apply for an affiliate program and receive your API access.
Limitations. Booking will vet you. Some APIs may not be available and no more than 20 simultaneous API connections are allowed.
Use cases. Accommodation booking.
Very similar terms are available at Agoda Affiliate Program as they belong to Booking Holdings.

Expedia Rapid APIs
The APIs are also aimed at accommodations. As with Booking.com, you get an affiliate fee.
How? Apply for the affiliate program and receive your API access.
Limitations. Each API has its limits of updates per message. Consult the FAQ at each API documentation for specifics.
Use cases. Accommodation booking.

Ticketmaster APIs
Ticketmaster provides info about and ticket reservations for concerts and sport events. They have two open APIs that are free to use.
How? You have to create an account on the developer portal.
Limitations. Up to 5000 API calls per day.
Use cases. Tour and attraction services.
Other suggestions to try?

Techtalks: Traveltech

Other topics:

What is the best pharmaceutical API?