Hardcore is such a vast term…I personally am more into modern hardcore, not so much the old school or tough guy hardcore, but I do enjoy it as well.
Some of my favorite songs include “Drop out” by Converge, “Board up the house” by Genghis Tron, “Panasonic Youth” and “Sunshine the Werewolf” by the Dillinger Escape Plan, “The End of All Things Will be Televised” by Norma Jean, but perhaps these are a bit too hard for your taste.
If you are more into metalcore (like Avenged Sevenfold), I recommend these bands:
Inhale/Exhale
As I Lay Dying
August Burns Red
If you are up for some Southern sound, I highly recommend Maylene and the Sons of Disaster.
If you enjoy post-rock, try Callisto or God is An Astronaut. If you enjoy post-hardcore, try out Battle of Mice, Khoma, Impure Wilhelmina, The Ocean, or Rosetta.
Finally, if you’d like to try experimental hardcore, you can check out Exotic Animal Petting Zoo, Psykup, or better yet: Fantomas!
Start with this epic song: http://www.youtube.com/watch?v=M8K8K1HEjV0 ;-)