Argh. I wrote a whole lot, but just do this: go here:
http://en.wikipedia.org/wiki/Electronic_music and read up, exploring all the subgenres and artist pages. You'll learn a lot more that any reply you'll get here (not trying to be dismissive of the replies or the posters here, just that a message board probably isn't the best way to go).
And for my personal picks...
Listen to
Squarepusher - exciting, often melodic Drum n' Bassy stuff, some people qualify it as IDM, which stands for (hilariously enough) "Intelligent Dance Music".
Throbbing Gristle - a seminal electronic group from the 70s
Psychic TV - lead by the singer from Throbbing Gristle, made an AMAZING "acid house" record called
Towards The Infinite Beat, big in the late 80s and early 90s. Currently touring under the name PTV3.
808 State - Techno/Acid House act from Manchester, big in the scene that was happening up there late 80s and early 90s.
Boards of Canada - Would be classed as IDM, sort of ambient like. Did a great album in 1998 called
Music Has the Right to Children which was pretty influential.
!!! - I
love !!! (can be pronounced "chik chik chik") I hear it described as "disco punk" but I'd it's so, so much more than that. I'm trying to think of similar artists but they have a pretty unique sound. Their song "Me and Guliani Down by the Schoolyard" is incredible, and I personally crown them the most exciting act around today, music you can dance and sing along to (although feel free to do the former, avoid the latter - at least when in good company). On the same record label as Boards of Canada and Squarepusher.
Dead Voices on Air - I guess it's classed as IDM. The guy behind it, Mark Spybey is a psychiatrist by day, curiously enough. They haven't released anything for a while, but their "Frankie Pett Presents" out on Invisible Records is great.
Of course there are tons of other great artists, but I honestly can't be bothered listing them, and I doubt it contributes anything worthwhile to do so.
Avoid
Bad House
Trance
Trance
Trance
and Trance