1. Enable the plugin
Before you create your first Groovy application make sure Groovy plugin is enabled in Settings → Plugins.
2. Create a new project
Open Project Wizard and select Groovy template. Since Groovy requires Java you have to specify the Project SDK.
If you create a Groovy project for the first time IntelliJ IDEA will offer you to create Groovy SDK library. Press Create button and choose directory with a Groovy SDK.
The IDE will create an empty project.
3. Create a new class
The easiest way to create Groovy class or script is to use Ctrl + N shortcut from Project View or Navigation Bar.
Choose between class, interface, enum and annotation with Up and Down arrows.
Let's create a class with a method returning "Hello, world!" string.
4. Create a new script
Now we can create a script file via Ctrl + N shortcut.
Choose between script and GroovyDSL script with Up and Down arrows.
Now we can create an instance of our class and invoke hello method.
5. Run the project
In order to run the application you can manually create a Run configuration via Run → Edit configurations or run the active script automatically by pressing Ctrl + Shift + F10 shortcut.