My Approach is below:
In order to explore any API, first of all we should have basic understanding of API’s and its testing
Know about the type of web services specifically, is it SOAP or REST
You should be familiar with XML, JSON formats and syntax.
What is the structure of the message like header, body, parameter etc.
Then should know some of the clients like PostMan, and Advanced REST client for Chrome and how it works
You should know what is a API Request and What should be there in the response?
What are the HTTP methods current service/API supports - like GET, PUT, POST, and DELETE.
Most important is knowing how to configure a request in PostMan/ Ad RestClient
You should know about list of HTTP status codes and what it mean.
After that Explore given service:
Read/ Explore the documentation if it is available and try to understand.
Know the endpoint, and what are the operations it supports
After getting basic idea
Try to call the service by carefully looking for what are inputs required to make a proper call, and check the returning results.
Understand how the invalid requests and exceptions has been handled.