My search for new sounds to listen to has led me back to Epitonic.com, a place I visited years ago when NewMusicBox wrote an article on their inclusion of 20th (21st?) Century Composers. In any case, I was poking around and realized that the best way to be introduced to new music is through their radio feature. You can choose as many "genres" as you want from a list of dozens of fun choices (for example Dronology, No Wave, Math Rock, Power Pop, Shoegazer, Slo-core, Noise, etc.) and choose the number of tracks you want to listen to (20, 50, 100, or 500) and it creates a custom-made, random playlist.

I am having fun.

My current playlist starts with these songs:
The Bees - A Minha Menina (a cover of a great Mutantes song)
Quasi - Mammon (poppy, melodic indie rockers)
Windy and Carl - Trembling (a long, trippy drone that, um, well, trembles)
Jad Fair and Daniel Johnston - Undying Love (strange, strange microtonal music, with a singer that lisps)
Glenn Branca - Lesson No. 2 (avant-garde guitar God goes crazy)
Les Savy Fac - Wake Up! (punk, with plenty of accusatory screaming)

I highly recommend trying this. The more adventurous you are, the more satisfying it will be. But for the more conservative amongst the masses, they do have less colorful genres like funk, rock, and pop available.