I finally managed to get Search Engine Friendly (SEF) URLs working on the site. I did some research and found one of the Joomla developers recommending OpenSEF, so I downloaded the component as a ZIP file and uploaded / installed using the Installers-->Components administrator UI. That was straightforward. I'd never been able to get the built-in SEF to work using the Global Configuration UI and discovered that my .htaccess file was empty. I poked around for awhile and finally found where to download the htacces.txt file (which I renamed to .htaccess to get it to function under Apache). This file needs to be in the root directory of my installation.

Next I needed to enable both SEF in the Global Configuration and in the OpenSEF Component. Still no luck; I was getting 404 errors. Then I looked closely at the .htaccess which has a section that needed to be un-commented to get third-party SEF to work. I did that and it all looks good now.

One annoyance: the site that officially supports this component (www.open-sef.org) appears to be non-functional. This means the online manual and help are inaccessible. Otherwise, I'd say from a quick look, that OpenSEF looks pretty powerful. 

To recap:

1. download htaccess.txt
2. install the file in the root directory
   
3. rename it to .htaccess
4. use Global Configuration to turn on SEF
5. download OpenSEF .ZIP file
6. use Installers-->Components to upload the .ZIP and install
7. use Components-->OpenSEF to enable OpenSEF
8. un-comment the third party section of .htaccess, toward the bottom