Weezer and The Cure are NOT emo at all, but they have influenced a lot of emo bands.
Embrace- get the whole damn album, but make sure you get the remastered version. key songs I guess are "I Wish I" and "Give Me Back"
Rites Of Spring- also, get the whole album. a few key songs are "
Spring," and "All There Is"
those are the only two you need to whole album of. Now onto other songs...
Fugazi- while they are awesome, they aren't really that emo. But if you have to hear ONE song, make it "Waiting Room." Features members of both Rites Of Spring and Embrace, which is really the only reason anyone considers them emo.
Hot Water Music- 220 Years, Driving Home, Rooftops, Us & Chuck... shit man, I love HWM so much. I could probably make an entire mix of HWM. Everything they've done is good, but if you want classic examples of emo, avoid their stuff on Epitaph.
Paint It Black- Ghosts
Turning Point- (these four songs are essential) Broken, Thursday, Behind This Wall, Anxiety Asking.
that'll give you a start, although it is an admittedly pretty narrow view of the genre. However, I believe it shall lay a solid foundation. You should be able to figure out the rest from there.