I'm not new to Windows Azure, but I would definitely like to learn more. Really, the landscape has changed a lot in the last year or so, making "knowing Azure" about as complex as "knowing Microsoft" was about 10-15 years ago. Let's start off with some definitions.
X as a Service
"Software as a Service" (SaaS), "Platform as a Service" (PaaS), "Infrastructure as a Service" (IaaS); the computing world is absolutely going the way of subscription "services" - the cloud. So what does Windows Azure offer? Well, if you count Office 365, all of them. If not, then it offers both PaaS and IaaS, with IaaS receiving the most press, recently, as it fills a gap in more easily allowing "Private Cloud" users to migrate to the "Public Cloud"
How Do I Start?
Well, first you'll need a Windows Azure subscription. At the time of this writing, you can get a free 90-day subscription at http://www.windowsazure.com/
Next, you'll need something to deploy software to Windows Azure with. The easiest to get started with is Visual Studio 2012 at http://www.microsoft.com/visualstudio/en-us.