Microsoft provides two ways of learning about Azure Services:

1. Azure Services Traning Kit

The Azure Services Training Kit which includes a comprehensive set of technical content including hands-on labs, presentations, and demos that are designed to help you learn how to use the Azure Services Platform. The February release includes the following updates:

  • 19 demo scripts that walkthrough several of the services
  • 10 presentations covering the entire Azure Services Platform
  • 3 additional hands-on labs for Live Services'
The technical content covers services including: Windows Azure, .NET Services, SQL Services, and Live Services.

It can be downloaded from here.

2. "How Do I?" videos for Azure Services Platform

For Windows Azure:

How Do I: Get Started Developing on Windows Azure? If you're a developer and you're new to Windows Azure, start here! You'll see what you need to download and install, and how to create a simple "Hello World" Windows Azure application.
How Do I: Deploy a Windows Azure Application? In this brief screencast you'll see what it takes to move your application into the cloud – you'll see how to request and register a token, how to upload your Windows Azure application and how to move it between staging and production in the cloud.
How Do I: Store Blobs in Windows Azure Storage? In this brief screencast, learn how to leverage Windows Azure storage to store data as blobs. You'll learn about blob storage, containers and the API that makes it easy to manage everything from managed code.
How Do I: Leverage Queues in Windows Azure? In this screencast, learn how to use queues to facilitate communication between Web and Worker roles in Windows Azure.
Debugging Tips for Windows Azure Applications The Windows Azure SDK provides a development fabric that provides a "cloud on your desktop." In this screencast, learn how to debug your Windows Azure applications in this environment.
For .NET Services:

How Do I: Get Started with .NET Services? .NET Services are a set of highly scalable building blocks for programming in the cloud. In this brief screencast, you'll learn about the registration process, the SDK and the built-in samples - everything you need to know in order to get started.
How Do I: Harness the Microsoft .NET Service Bus? The .NET Service Bus makes it easy to access your Web services no matter where they are. In this brief screencast, you'll see how to take a basic Windows Communication Foundation (WCF) service and expose it to the Internet with the .NET Service Bus.
For Live Services:

How Do I: Get Started with the Live Framework? If you are looking to get started developing with the Live Framework, this is the place to start! In this screencast you'll learn how to get a Live Services token and what you need to download in order to start writing Live Framework applications.
How Do I: Use the Microsoft Live Framework Resource Browser? The Live Framework Resource Model is a simple, straightforward information model based on entities, collections and relationships. In this brief screencast you'll learn how to navigate the relationships between entities by using the Live Framework Resource Browser, which is a tool that ships with the Live Framework SDK.