This walkthrough covers what I found to be the simplest way to get a sample up and running on Windows Azure that uses the Table Storage Service. It is not trying to be comprehensive or trying to dive deep in the technology, it just serves as an introduction to how the Table Storage Service works.
Please take the Quick Lap Around the Tools before doing this walkthrough.
Note: Some of the code is cut off due to formatting however the code for this walkthrough is attached to this blog post.
After you have completed this walkthrough, you will have a Web Role that is a simple ASP.Net Web Application that shows a list of Contacts and allows you to add to and delete from that list. Each contact will have simplified information: just a name and an address (both strings).