What do you consider when testing software, particularly APIs?

What do you consider when testing an API? Whether/how to automate it, what you're actually testing, whether the system's state is relevant and such.I put together a blog post with my thought process applied to a simple API.

TLDR: The more consideration you give to what you want to test and how it's represented in the system, the easier the test implementation will be.

Let me know what's your view on that.