Is the API part of the product I’m testing?
Is the API a responsibility of the team I’m into or other teams’?
Is the API internal to the company or external?
What role am I in at the time? a tester, a developer, analyst, product manager…
What resources do I have available? do I have access to the API, how much, when, where? how can I check it?
How much time I have to test?
How often is the API updated, by whom, do I know when and what gets updated?
Do I have access to the code of the API? developers that implement it?
Is the API used, or will it ever be used?
What’s the API used for? What’s the risks that could trigger by using it?
What perspective do I have for the API? consumer or provider?
Who, what, how do other services/products need to use the API? does it provide(adjust) the data, and make accessible the functions that are mandatory for the product to be minimally integrated?
What’s the API type, how small it is?
What state is it in? planning, brainstorming, diagram, specification, coding?
Where does the API fit in the product context? How many clients are using it? What’s the SLA for it?
Who’s the owner of it, what does the owner want/need in regards to the information obtained from testing?
What is my mission? A mission is agreed with a product owner usually or your manager on what information about the product is needed for them to be confident in a branch merge, or a release or an integration or certification or performance or availability or localization or something else…
And on and on it goes the list of questions you should find answers to…
Other references(several posts): Exploratory Testing on an API? (Part 1) « Developsense Blog
Do this first and then once you are confident on the things that comprise the API context and you are aware of stable outcomes, you can automate checks as well if time allows and they are worth the effort.