You have a few options for getting started with the Google Apps Groups Settings 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 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.
If you already understand the basics and there are particular things you want to do, see the Developer's Guides such as the Prerequisites, Authorized Requests, Usage Limits and Manage Groups Settings.
To look up a particular resource type or method, see the Reference.