How do I find my passwords on this Mac Pro?

If and only if you are an Admin user, you can use keychain access to look up ONE password at a time. It is deliberately tedious, because one does not generally need to know more than one password at a time.

To gain access, select the password from the list in Keychain access and choose getInfo. check the [√] show password box, and a dialog comes up asking for your Admin password. Enter your Admin password and the password you requested will be shown.