API can be tested with three ways:
a. Using curl
b. Using a firefox plugin like RESTClient
c. Writing actual code in some programming language and sending data through that programming language.
Once you will get its feel, it is not rocket science. Just keep in mind there are two kind of API calls.
1. GET call when we ask server to send us some data. This is very easy to test in browser. You just visit a particular url and get data in XML or JSON format.
2. POST call when we send data to server in XML or JSON format and ask it to process it. This becomes little tricky as we have to prepare our data in XML or JSON format and then send it to server. This sending can be done via curl command line for testing, RESTclient plugin for testing or some real programming language.