I discover music a load of different ways, though the most prevalent is through other people's recommendations. For instance, on another forum, a member's username is Mogwai, and I was curious as to what it meant. He told me that it was the Scottish post-rock band. I tried it out (via Youtube, since internet radios suck and don't play post-rock music right o h m y g o d) and loved it, and was, from that point, a fan of post-rock music with the discovery of other bands in the same genre, etc.
I never discover new genres/artists via Youtube because when I use Youtube I end up watching specific video series' or listening to specific artists/watching specific videos. Honestly, I can't think of a time I've legitimately browsed the site for new things. I'm usually referred to things or I happen to stumble upon a genre/artist on a site or wiki.
I usually discover new music through the radio. If I don't know the name of the particular song i like, i write down a few lyrics and then google them. The song usually comes up in the first five or so results. Friends also give me new music to listen to, not to mention I browse Youtube for the latest hits.
Either via the radio, though the playlists are somewhat limited/repetitive, or via an app on my phone called Deezer which essentially allows you to search/play/download any song that you want for free with multiple ways of finding random songs that you've never heard of before or countless songs from a single artist and having them all play at once. I've ended up finding so many great songs and artists this way so now I've developed a habit of leaving the app to choose any random song it can find and there's a fair few gems amongst them. The only other way I tend to come across music is from friends/siblings playing it, but most of my friends like the same kind of stuff as me so that's also pretty limited.