>research the artist's history
Contextualizing the album can be very important. Looking up their personal history can give you an idea of what the artist is feeling and looking up how they fit in overall music history can help you appreciate what's innovative about the album for its time.

>read the lyrics
I think it's a bit of a mistake for people to ignore lyrics. If the artist put thought into writing them, then you should give thought into listening to them. Of course some artists don't have the most meaningful lyrics, but others can add a nice story to the song which makes it a lot more emotional and meaningful.

>look up music videos or concert performances
Sometimes a visual component is very important to an artist's style or message. It can add something extra to the music. Even if it isn't that important, it's still nice to see how they do things live.

>give things multiple listens
This seems kind of obvious, but I'm still shocked as to how many people don't do this. And no, this isn't about "forcing" yourself to like something. It's more about recognizing the whole shape to a song or an album. You can recognize things like if a long song has a recurring melody you won't notice on the first time, or if it's a concept album and there are repeating melodies across different songs. Also, it can make you more familiar with a melody so once you're done listening to a song for that, you can listen to and appreciate other aspects in the song as well.

>research music theory
Don't be intimidated! Some of it can be very complicated, but you don't have to memorize all the different sorts of scales and chords, and you don't have to learn what the difference between a D and a C note is. Just going into the basics can give you a nice vocabulary for you to discuss music better. If you know what dissonance means, or what a modulation is, you can be more aware of it in a song and its effects will be more effective on you as a listener.