Shazam is a great app for finding out what song is playing. Good for when watching TV & movies and you hear a song you like (just tap the "listen" button and it almost always will be able to identify the song).
On Spotify, Discover Weekly and Release Radar are playlists that are automatically made for you (weekly) based on the music you listen to. I almost always find at least a few new songs per week that I save from those playlists. Spotify is also good for simply searching whatever genres you are into and exploring different playlists (I usually search the genre name then look for a playlist that already has a few songs/ artists that I enjoy on it, so it's more likely that the playlist is catered to my taste).
Spotify aside, I've recently gotten into a habit of simply writing down an artists name in my phone notes whenever I am listening to a song that I like (if I think there could be more songs I would enjoy by that artist), whether it be a new song I've just heard, or even an old song I've been listening to for years but still haven't ever gotten around to listening to more of the specific artist... So I always have a list of artists to explore, and now whenever I have time to listen to music I will explore the discographies of those artists, and always end up finding some good music I haven't heard before. I said "Spotify aside", but I do usually use that app when exploring albums, since it's so convenient and Spotify generally has most of any given artists albums. You can also simply search an artist and just check their top 5/top 10 most popular songs if you don't have time or patience to look through each album.
Last.FM is also a great site for exploring artists. If you search your favourite artists on that site there is a "similar artists" section that brings up a list of artists, its rates how similar they are musically to the artist you're looking at, shows their popularity and gives a summary of each artist and their musical style which helps a lot when looking for someone new to listen to.
It helps a lot to have certain times/activities that you routinely use to incorporate music in, so you are more likely to spend time listening to the albums/playlists. For me it's during driving, while running/exercising, playing certain video games, or just while sitting around/having a meal. The mixture of the things I've mentioned above has created an endless cycle of new music for me to discover.