If you are not already familiar with the Arctic Monkeys, start out with the album Favourite Worst Nightmare, followed by Whatever People Say I Am, That’s What I’m Not. These two albums are fantastic, in my opinion; they also serve as a great foundation for getting to know the Arctic Monkeys.
Another band that is fantastic (and underrated) is Cage the Elephant. I highly recommend checking out Thank You, Happy Birthday and their newest album, Melophobia. I think you would like the song “Aberdeen” — the song’s name is an allusion to the birthplace of Kurt Cobain. Also, Cage the Elephant have toured with The Foo Fighters (the lead singer is Dave Grohl, who was in his early days the drummer in Nirvana).
It makes me so happy to know that you like MGMT. I am a huge fan, and I would recommend listening to Tame Impala (similarly psychedelic), Pink Floyd (similarly experimental), Led Zeppelin (MGMT said in an interview that they aspire to emulate this band), and The Flaming Lips (MGMT has been influenced by them).
Now I feel like listing some bands that I feel you would enjoy.
The Doors, The Violent Femmes, Neutral Milk Hotel, Vampire Weekend (THEY ARE THE LIGHT OF MY LIFE, I KID YOU NOT), Interpol, Modest Mouse, Vance Joy, The Strokes (!!!), The Killers, The Smiths, The Cure, Pearl Jam, and The Offspring.
I have so many more suggestions, please let me know if you want more/more specific music-listening advice!