You should probably check out the Substance albums. They give a pretty good overview (though some of the songs are longer, re-mixes, I believe). But if you just want some specific songs, then here are some of my favs:
1. "Love Vigilantes" off of Low-Life
2. "Touched By The Hand Of God" - Singles
3. "Thieves Like Us" - Substance/"Pretty In Pink" Soundtrack
4. "Ceremony" - Movement or Substance
5. "Temptation" - Substance
6. "Shellshock" - Substance/"Pretty In Pink" Soundtrack
7. "Round and Round" - Technique
8. "Perfect Kiss" - Low-Life
9. "Bizarre Love Triangle" - Brotherhood
10. "1963" - Substance (Disc 2)
11. "Lonesome Tonight" - Substance (Disc 2)
12. "Crystal" - Get Ready
I know "Blue Monday" was their biggest hit, but that song was SO overplayed that I just got sick of it and it just does nothing for me now.
Listen and enjoy. New Order are awesome and essential for a proper 80s background!