If you want to get started with Azure, you might be interested in listening to the 4 last episodes of ASP.NET Podcast, by Wally.
Episode 131 is a general discussion about Cloud Computing in general (MP3 only).
Then things get more interesting, and Wally talks about two of the main components of Windows Azure: Blob Storage and Table Storage. These two episodes are screencasts that show how to setup the development environment, how to use the local Azure fabric and the how to create a simple CRUD application.
Then the last episode (#134) is the a wrap up show, with general thought on Azure development.
If want a more complete sample, that shows how to deploy you app on the cloud you might want to check the great series by Maarten Balliauw: CarTrackr on Windows Azure