Hit Charts
The first, and probably most common way to discover new music is by looking at lists of current top songs. This shows what everyone is listening to right now and what they are loving. Some of the most common charts you can look at are the Billboard Top 100, iTunes most popular songs, and Spotify's top songs. I like to look at music that is currently popular in other countries (some of my favorites are England, Australia, and the Czech Republic) because music from other areas of the world can be really different than what is popular in your own country, and can give you a bigger variety of songs to listen to.
Pandora Radio
Another popular way of discovering new music is through Pandora radio (or apps like it). You can look for certain genres of music, artists that are similar to your favorites, and top songs in your area. This has been my way of getting new music for a long time, and I have had a lot of success with discovering new music to listen to. One of my favorite Pandora stations to listen to is Lorde Radio. Most of the music on that station has a really unique feel, including a lot of synthpop and some pop punk songs.
Spotify
My favorite way of discovering new music is through Spotify. In case you don't know what that is, Spotify is a free music app that you can download on your computer or phone. You can look up mostly any song that you want to listen to, and you can also make playlists that people can follow. The social networking part of it is helpful because you can see what your friends are listening to, which can help you discover new songs. There is a section of spotify that is full of public playlists that you can listen to. There are playlists for almost every mood, situation, genre, time period, and age group. There are also always up and coming artists that promote their music here.
Blogs
If you want to find music that people online love and that might not be well known to everyone, blogs are the way to go. If you search for music blogs or popular blogs on Google, you can probably find tons of blogs that frequently post about music they are currently listening to. They also provide their opinions on the songs and sometimes give detailed descriptions of what the music is like. There is one blogger I go to a lot for new songs to listen to, and I have not been disappointed yet. And hey, this is also a blog that you can go to to discover new music.
Google Search/Youtube
This one might seem kind of obvious for some people, but Google search is also a great way to find new music. It might be kind of tedious and hard to find songs that you think are good at first, but with some patience it will eventually become easier to find what you are looking for. Also, if you go to Youtube and listen to a song you like, you can find suggestions for other songs on the right side of the page.
Some of My Current Favorite Songs
5 Years Time-Noah and the Whale
Applause-Lady Gaga
Earned It-The Weeknd
East of Eden-Zella Day
Elastic Heart-Sia
Empty Gold-Halsey
Hideaway-Kiesza
Home-Edward Sharpe and the Magnetic Zeros
Immortal-Marina and the Diamonds
Lost In the World-Kanye West, Bon Iver
Me-The 1975
Night Changes-One Direction
Riptide-Vance Joy
Run-Daughter
Sweet Ophelia-Zella Day
The Driver-Bastille
The Wire-Haim
Ugly Heart-GRL
You're Mine-Lea Michele
Four Five Seconds-Rihanna, Kanye West, Paul McCartney
Firestone-Kygo, Conrad
Flowers In Your Hair-The Lumineers
Hold Back The River-James Bay
I Choose You-Sara Bareilles
I Lived-OneRepublic
Run Daddy Run-Miranda Lambert, Pistol Annies
Your Bones-Of Monsters and Men
Into the Lair-Zedd