Here's the obligatory Project Euler link, which is a petty universal means of familiarizing oneself with a new language. The problems are not ideal for learning the more macro aspects of programming ( OOP, Design Patterns, etc), but it will give you a good feel for Groovy's syntax and basic features. I've ground through just shy of 100 of them, with definite results. - Plus, it was fun.

I'd also recommend the Groovy Web Console. You can use it to run your scripts online, but perhaps more importantly, you can browse the scripts of others - I've found this to be rather invaluable, particularly while getting acquainted with Groovy's DSL capabilities.