I'm looking for a website or websites that have free (or cheap)
databases available for download.  For example, Microsoft.com has a
"templates" section where prebuilt databases can be easilly downloaded
but I was hoping to find more variety elsewhere.
Databases that relate to logging customer or employee information
would be ideal as well as information for Call centers such as
databases that track call statistics, etc.
The next part to the question is...what would the best resources be
(websites, books, etc.) on how to learn about database design?

I encourage you to search for www.dotnetnuke.com, log in and download
the portal version 2. It is based on access database, (from 3 it is
designed to use SQL Server) and it is smartly renamed from .mdb to
.resources to avoid database download from your portal.

U can use phpMyAdmin ( MySql ). It is open source and free of cost...
Plus it has got many functionalities....Or you can have MsAccess as
second option....