When I first started using Stack Exchange, I did not have an e-mail account of my own and was using my father's. Now, I have my own e-mail address, but I can not change the address that I use to log in. Is this currently not possible or is it possible but I have not yet figured out how? I am using a Stack Exchange OpenID.
No, it is not possible to change the email of your Stack Exchange Open ID account. As far as Stack Exchange OpenID is concerned, email is your identity. You can't suddenly be someone else.
What you'll need to do is go to your profile and click "my logins". This dialog will allow you to add/remove as many login credentials as you want. From there, add a new Stack Exchange credential - setting up a new SEID account.
Once you have the new account set up and connected you can remove the old account.