My tastes in music are schizophrenic, however I tend to gravitate to either EBM/synth etc or Shoegaze.
(If you have never heard of this genre, I have included a WIKI page to feed your curiosity
http://en.wikipedia.org/wiki/Shoegazing )
Off the top of my head some of my favorite Shoegazer bands include
Slowdive, My Bloody Valentine, Love Spirals (formerly known as Love Spirals Downward) M83, Mogwai, (some people classify Radiohead (from the album OK Computer forward) as Shoegaze/experimental), Lush, Cocteau Twins and Sigur Ros.
Groups like The Smiths, Ride, Suede(formerly known as The London Suede) , Mansun, The Telescopes also fit into this genre/have roots in Shoegaze.
A few songs I would recommend to get started would be
Slowdive: "Ballad for Sister Sue"and "Primal" from the album 'Just for a Day',
and "Alison" and "When the Sun Hits" from the album 'Souvlaki'.
Radiohead: "How to Disappear Completely" from the album 'Kid A'
Love Spirals Downward "Will You Fade" from the Album 'Ardor' , "Love's Labour's Lost" from the album 'Idylls', and one of my favorites, "I could only find it by Chance" from the album 'Ardor'.
Also, My Bloody Valentine's album "Loveless" is just... amazing, and if you read about the sound engineering that Kevin Shields did to produce this album you will be aweinspired.
Anyways, it's not for everyone, but as for me my playlists usually go back and forth from EBM/synth to Shoegaze.
I hope you found this informative. =D