Specifically, is sharing and downloading MP3 files (or other types of music files) and videos illegal?
It is true that some copyright holders give official permission to download MP3 files and you might be able to find a limited number of videos that are not copyright protected. It is also true that some MP3 files are copyright free and some MP3 files can be legally obtained through subscription services. However, most MP3 and video files that are shared do not fall into any of these categories. US Copyright Law allows you to create MP3s only for the songs to which you already have rights; that usually means you purchased the CD or downloaded a file via a subscription service. US Copyright Law allows you to make a copy of a purchased file only for your personal use. Personal use does not mean that you can give a copy to other people, or sell a copy of it.