As some users have suggested below, Fabrizio De André is probably the best songwriter to listen to if you want to enhance your Italian. Although the meaning of some songs is hard to understand and some others are in Genovese, most of his lyrics are quite clear and his Italian is excellent. There's a page which collect explanations of the songs, links to youtube videos and translations in English. Search for the songs he sang together with the PFM, they are a good start:
http://deandretranslated.blogspot.com/
These are posts I wrote about Italian songs of the Sixties:
https://www.italki.com/discussion/198274
and of songs for children (which are quite easy to learn):
https://www.italki.com/discussion/216069
To add other songs I've listened to recently, I would suggest these:
https://www.youtube.com/watch?v=VSAB5ckdCv0
https://www.youtube.com/watch?v=Ej0ME8xdiF8
https://www.youtube.com/watch?v=gF1XRQGhe8s
(the idea of the video isn't that bad but it looks like a big advertisement for sportswear)
https://www.youtube.com/watch?v=RZO9F0f811c
https://www.youtube.com/watch?v=0Txpx6D1K6Q
https://www.youtube.com/watch?v=Tpehv8IC2tA
https://www.youtube.com/watch?v=WEOGrMVRDXk