So I'm buying a $5 .net domain name from Register.com. Their renewal is $38, so my plan was to let it expire, then when it expires I will register to it again, but with HostGator or something for $15. Is that how it works? I'm kind of nervous about it, because I don't want Register.com to register the domain after it expires, and make me pay $40 for it.
Register.com may have an auto-renew option. Turn it off in the control panel.
Transfer the domain to another registrar before it expires. You would go to your new registrar's control panel (HostGator...), pay a for new year, and initiate the transfer from that control panel.