Since my teenage years, I've been listening almost exclusively to Metal at home (of course I hear other kinds of music on radio).
When I want a change from Metal or something softer, I usually listen to Blink 182 or Boxcar Racer.
Starting to like Angels & Airwaves.

Some U2, Journey, Styx, etc can also do.
Unlike lots of Metalheads though, I hate hard rock in general.

Tons of bands...
I love The Byrds and The Stones...
That said, i will keep it in the music i listened to the most when i was a teenager back in the 80's (non-metal bands from the 80's):
Sonic Youth, Pixies, REM, Dinosaur Jr, The Jesus and Mary Chain, Joy Division, The Cure, The Cult, The Church, Galaxie 500, The Stone Roses, The Sisters Of Mercy, The Long Ryders, Dream Sindicate, Violent Femmes etc
At that time i didn't, but i must say that now i think New Order were also great.

I second Sonic Youth, they are the shit. I like Wilco, Radiohead, REM, and even the most recent Opeth because that is not metal, but i like it. Some killer Allman Brothers type guitar duels in that, and I dig the guitar duel kinda thing.

Given I like guitar in general; Frank Zappa when he's not talking about finger-banging his sister, Junior Brown is a pretty awesome country slide player, Doc Watson is awesome as well.

Plus Classical and Jazz. I either absolutely love both genres or find them utterly shit. There is no in-between on Classical or Jazz stuff. It either sucks or I want everyone in a three-mile radius to shut the fuck up so I can enjoy it!!!

I don't really consider myself a massive metalhead (though the mirror would disagree with "not massive" ), so I listen to quite many non-metal bands. I love Radiohead for example. I think Thom Yorke and Johnny Greenwood are right up there with the greatest songwriters of all time and somehow - with the exception of "Pablo Honey", Radiohead have never released a bad or mediocre album. I think after "Creep" became a hit and after "OK Computer" won them all the accolades, they could have sold out and cashed in so big, but they went in the opposite direction and forever solidified their status as an all-time great band with Kid A.

My two favorite genres in addition to metal are prog and punk rock. I almost never listen to either genre on the same day or week, however. Both genres sort of come and go. Some weeks I will listen to nothing but Pink Floyd, King Crimson, Rush, Spock's Beard etc and another week I will only listen to The Sex Pistols, The Clash and some punk bands from my native country, such as Vennaskond, J.M.K.E. and Singer Vinger.

And also, I absolutely love David Bowie.

Nhors non metal albums are amazing!

Ever since I kicked my issues with depression and anxiousness I've been moving further and further away from the standard metal music, that is my comfort zone. From what I understand it has something do with trying to find comfort and distraction when you're feeling sick or something, so now it's all about curiosity and challenge.

Some of the new non-metal bands that lately got me hooked are Sparks, The Cure, Siouxsie and the Banshees, Young Marble Giants and Television. This whole New Wave and experimental rock scene from the late 70's and early 80's was almost completely alien to me until recently, and now I have fallen in love with it. Would you believe I never even bothered to listen to The Cure, one of the best bands ever, until just recently? They're awesome, especially Disintegration, their late masterpiece with that pitch perfect production.

Sparks is a whole another kind of beast, like these guys do and say just whatever they want and the end result's most the time highly interesting and creative to say the least. My first exposure to their art was an album called "No. 1 in Heaven" where they basically cut the glam rock thing and worked with Giorgio Moroder to produce some kind of experimental disco rock. It's amazingly fun and unique album.

Edit. Oh, yeah! Tears for Fears, sometimes considered progressive pop for being such a subtle, unique and smart band. Their first two albums are superb, the debut being a concept album about abuse and depression.

Mille Monarques
Very obscure but surprisingly great Quebecois band I came across on YouTube (strangely enough, theres a version of Amorphis's "To Father's Cabin" that's falsely linked to one of Mille Monarques's songs). They play very chill, catchy, proggy post-grunge. They only ever released one album (self-titled), apparently, and the album is hard to find. On Bandcamp, the songs are all randomly shortened for some reason. On YouTube there's a playlist of the full songs: https://youtube.com/playlist?list=OLAK5uy_mOv-RPbXiAhYbcmcQOrZDGLxXYx68CilY&feature=shared

Here's a taste: