I started seriously listening to music as a younger kid in the 70' where my first albums were Queen and Kiss albums. Kiss obviously had the marketing, Queen just had the amazing hits
I was a teenager throughout most of the eighties and spent most of that time listening to metal, punk, and alternative stuff as well as discovering bands I had missed out on from the 70's like Led Zeppelin, T.Rex and Sabbath etc...
Once in a while I'll hear something on radio that catches me.
Other times I'll actually wade through the iTunes or Amazon "other people also bought" lists. I've found quite a few gems in there. I discovered the Yeah Yeah Yeahs that way and they became one of my favorites.
Going to shows at the various local music venues. I've caught many bands who made it big this way before they made it big
as other have mentioned, you tube and Pandora are nice.
speaking of more electronic type music that still rocks...
I've found tv commercials sometimes offer up some hidden gems as well. I've came across Ladyhawke
https://www.youtube.com/watch?v=p1Pucpi8cvo (recently in Verizon commercials) this way, Sleigh Bells (from the apple commerical)
https://www.youtube.com/watch?v=nmFgejWZjtg and Goldfrapp
https://www.youtube.com/watch?v=uco-2V4ytYQ (from some phone commercial ). Basically just google a few of the lyrics I found, then it usually leads to a songs lyrics site, then check out the song on you tube to make sure its the right one
as far as some of my favorite more modern albums and bands of the last decade, my favorite being...
The Flaming Lips. Soft Bulletin and Yoshimi and even At War With The Mystics was a trilogy of releases that were just sonically so far ahead of what most bands were doing. It's like you take guys who started out as basically punk rockers, grew up, introduce them to pink floyd and neil young and write some songs together.
For me it was the equivalent of the Beatles around Sgt Pepper era as far as experimenting with sounds, but still keeping an almost pop like catchy sound underneath. They did a remix of each of these albums in 5.1 surround sound that is just amazing to listen to the depth of everything going on. I'm surprised they never broke mainstream besides a minor hit or two. One of the most underrated bands ever.
Foxy Shazam - A singer who reminds me of Freddie Mercury reincarnated, and a band that grew up on 70's glam and arena rock with a good dose of the Stones who still sound modern at the same time
https://www.youtube.com/watch?v=4rz4I69mQMo
https://www.youtube.com/watch?v=OFt3OqmGSbI
Yeah Yeah Yeahs
Shiny Toy Guns
Wolfmother