Bruce

Thanks again, Ipstenu, for your advice. The hierachy page looked scary.

I eventually decided to use the multi-site capability of WordPress.
I rang my domain host and they quickly walked me through setting up a subdomain on my hosting account, rather than on CPanel as I had mistakenly done. We then pointed this to the server’s IP address.

Then I followed the instructions to ‘Create a Network’ on WordPress, and “hey Presto” it didn’t work!
The solution was to to disable, then re-enable, the WordPress theme.
I’m very happy with the result – two websites with two different looks that I operate from one WordPress installation.
Would recommend this to anyone.
Cheers, Bruce

Thanks, Ipstenu, for your advice about subdomains and subfolders.
Exploring the simpler option for the moment, I’m interested in your comment,

3. Possibly, but that would just be theme settings, and you said you tried some of that. If you just want different pages to look different, you can use page templates and so on to change the design. I don’t know if that’s ‘easier’ or not for you.

I’ve found it’s possible to change the basic layout for each page by selecting the default, side bar or showcase templates (offered in Twenty-Eleven theme). But I’m not sure how I could use these templates to give a different look for each of the 3 page sets of my domain i.e. 3 different headers/fonts/colour schemes/page menus.
How would I use templates to achieve this?
Cheers,
Bruce