I've pointed it at the Google app servers, and unfortunately that means a redirect to
mail.google.com. Is there a way to avoid that redirect and allow me to browse email at
mail.example.com? Will I have to host my own mail server if I want that to happen?
It sounds like you want to have your domain name point to your Google Apps mail. If if that is correct, then add the following CNAME record to your domain's DNS settings:
Name/Host/Alias: mail TTL: 3600 Record Type: CNAME Value: ghs.googlehosted.com.
Here is more information from Google.