If I'm producing a commercial or a movie/tv show and have a specific song and artist that I want to play in the background of a scene or a montage, what are the rules for using that song? Who do I reach out to to get permission? How much does it cost?
You need a sync license [Music Synchronisation License].
You need permission from two different organisations - the one who administers the song rights & the one who has the recording rights.
Where you get it depends on what jurisdiction you live in & who operates the rights. For instance, in the UK you would first approach the PRS [Performing Right Society] who work in conjunction with the MCPS [Mechanical Copyright Protection Society].
Pricing is on a case-by-case basis.
General guide to sync rights - Wikipedia: Synchronization rights
To start, find who is responsible in your territory & see what their web site has to say about it.