To be a good music listener, you first need to understand one important fact: there is no good music or bad music. If all the musicians you are listening to are famous enough and they have a huge following, you should understand that they are famous because they play good music. Just because some musicians don’t match your criteria for good music doesn’t meant that they are bad producers. Therefore, listen to everything. Do not restrict yourself to a certain genre. You will come across several metal fans who will say rap and hip hop is not a good genre, and the rap and hip hop fans may state otherwise. Just try to get the best out of each and every genre you listen to.
Once you do not restrict yourself to a single criteria or genre, you will understand how different musicians are extremely good at what they do. For example, a young kid like Justin Bieber has a huge fan following and the reason is that he has been pleasing a lot of fans with his music. Those people like him for what he sings, and just because you do not like him, it does not mean that he has not been making a mark in the world of music.
Lastly, if you think that a certain genre is not pleasing you and is not made for you, simply give it up. Don’t say that it’s bad; just say that it is not your thing. There are others who like that genre and it does not mean that it is bad. Simply give up on that genre and become the master of those musical pieces which are pleasing to your ear. Listen to everything related to the genres you like - only then can you become the best music listener.