how do I permanently delete old mail or junk mail? It keeps coming back!

Have you tried Emptying the Trash after deletions? Or do you use the Trash option in Mail? If you do the latter then after the first deletion, go to Mail's Trash mailbox, select the content and delete again. Do not Quit Mail until everything is gone that you don't want.

If you open Mail's preferences, select your mail account. On the right click on the Accounts button in the toolbar. Now click on the tab labeled Mailbox Behaviors. The lower set of dropdown menus will let you set options for automatic removal of junk and Trash. Choose the most suitable option for you.

I would also recommend getting a good Spam filter. Mail's Junk filter is fairly good but SpamSieve is much better.