I haven't found a great need for the queue so far. Maybe it's just that I'm not seeing it in my app view. Could also be that the data you can store in the queue is minimal. You basically store short text strings (like record ids), and then you have to do something with the ID when you pull it from the queue, such as look it up, delete it, whatever.
In my app, I didn't use the queue at all, just as Peter suggests. I wrote directly to table storage (accessed via it's REST interface using StorageClient) from the client. If you want to look at a concrete example, take a look at http://www.netalerts.mobi/traffic. Like you, I wanted to learn Azure so I built a small web site.
There's a worker_role that wakes up every 60 seconds. Using one thread, it retrieves any new data from it's source (screen scraping a web page). New entries are stored directly in table storage (no need for a queue). Another thread deletes entries in table storage that are older than a specified threshold (there's no issue with running multiple threads against table storage). And then I'm working on the third thread which is designed to send notifications to handheld devices.
The app itself is a web_role, obviously.