Jules - thanks!
To host on your own domain, you need to use wordpress.org, not wordpress.com (which is the free hosted version).
My hosting company makes it easy with a quick click to automatically install to my root directory, but you can also do it manually. Just follow the instructions for the download at
wordpress.org.
[off-topic]
After I had written my first post and was driving to town, I started thinking I should clarify: I think web designers are the experts in web design, and others who use templates etc should respect the knowledge of the true designers.
If I were to take Wordpress training, it would not be to call myself a WP designer - because I'm not qualified to do major changes to templates or in-depth tweaks. I believe the professionals who "speak" html/css as a second language really should handle those types of tasks. However, knowing and understanding WP enough to make small changes to pre-designed templates in addition to basic set-up, installation and customization through plug-ins is perhaps a value-added service that I could add to my offerings and perhaps become a small niche for my business.