If I want to cover a song on a CD and release it to the public, how would I go about doing so (legally)?

Lol no they wont badger you they will sue you into oblivian. What you woudl have to do is get permission from and or pay the owner. If they have are not dead. If they are there is a certain number of years they have to be dead before you can use it, but in most cases the rights are willed to someone in the event of death and that is who you must find. Bottom line if you record and distribute someone elses property and collect money for it you will be sued for stealing. You can easily find the information by looking on the net for Copyright infringement. Hope that helps.