Speaking as an autist, one of the earliest bands I've ever got into was Pink Floyd at the age of 13, before then it was Daft Punk at age 12. They weren't necessarily appealing to me because of their popularity, they were just new to me and I really enjoyed listening to them. At 14, I discovered the infamous website RateYourMusic and from then on my musical horizons grew. I also had a teacher in my middle school's special needs class who was also into music and introduced me to Radiohead, Arcade Fire and Department of Eagles (this was around 2008-2009). I didn't care for Radiohead then, even though I did later on and don't care as much now.
As a 14-year-old I became disillusioned with the label "classic rock" as it moreso meant an indicator of age rather than quality. Some AOR bands like Foreigner, Boston, and Van Halen at that age sounded very commercial and cliched, and I felt they're only considered "classic rock" because they put out most of their stuff in the 70s and 80s. At the same time, there was great music during this time like Television, Pink Floyd, King Crimson, and even the more radio-friendly Cars, and it was kind of weird to put them in same place as bands like Kiss. My thread "Classic rock that sucks" kind of shook the little RateYourMusic community back in 2009, and it made me a very controversial figure on the site.
If you want to look for popular music that has a great focus on vocals and lyrical themes, I think the most orthodox choice is Radiohead, but I wouldn't recommend jumping into them immediately. They sound a bit colorless to virgin ears, so I suggest you take your time with them.
If you want emotional vocal performances and themes, but kind of accessible and easy to get into, I recommend Arcade Fire. "Funeral" is a great album if you're looking for that sort of popular music. It's debatable whether Arcade Fire should be considered "mainstream" or "indie," especially now, but millions upon millions of kids my age must have heard this album once in their life. Enjoy.