While it is quite easy to write and run API tests with Postman, as your API gets more complex and the API calls rely more and more on prior successful calls, you reach a point where having automated scripted tests is a much better option.
This video walks through building and running a collection of tests for an API.
The API used for this video is available in this repo:
https://github.com/prof3ssorSt3v3/postman-test
An exported copy of the tests used in the video is also included in repo.
Postman documentation:
variables - https://learning.postman.com/docs/postman/variables-and-environments/variables/
collections - https://learning.postman.com/docs/postman/collections/intro-to-collections/
tests - https://learning.postman.com/docs/postman/scripts/test-examples/
https://learning.postman.com/docs/writing-scripts/script-references/test-examples/
Intro to using Postman video - https://www.youtube.com/watch?v=t5n07Ybz7yI
Postman Mock Server video - https://www.youtube.com/watch?v=pAD11I3k9q0