You have a few options for getting started with the Google Calendar API.
The one-minute experience: APIs Explorer
To play around and see what the API can do, without writing any code, visit the APIs Explorer. Instant gratification!
The five-minute experience: Quickstart
If you're using a supported language, you can download a client library and run a small sample client application by using Quickstart:
The one-hour experience: Your First App
For a more detailed step-by-step tutorial explaining how to get up and running, follow the instructions in Your First App.
To look up a particular resource type or method, see the Reference.