Based on your choices there... Breaking Benjamin are awesome. Gonna ditto Three Days Grace, in particular the album One-X. Skillet, RED, and Flyleaf are pretty damn good - they're Christian but they don't shove it down your throat, and their songs are often absolutely nothing to do with their faith (or are vague enough so that it's not necessarily religious) so it really doesn't matter. Same goes for Spoken - their album "Illusion" is fantastic. Fireflight are all kinds of amazing, though I don't know their discography to the same extent.
I'll say Stone Sour as well, 'cause in my opinion they've got a slightly different and more mature sound. Headed by Corey Taylor, the guy who sings (growls?) for Slipknot. Turns out the guy can actually sing pretty well too. Try Enter Shikari if you feel like shaking things up - they're a bit "experimental", add some electronic aspects to their music, which is a nice change from the occasional repetitiveness of hard rock bands. Deftones are great too; pioneers in the genre, really, they've been around for a looonnnggg time. Chiodos, 30 Seconds to Mars, D.R.U.G.S, and Emery are also solid choices. Evanescence's Fallen album is golden, as is Halestorm's "The Strange Case Of..."
If you like HU, then go check out some of Deuce's solo work. Not my cup of tea, but he followed largely in the same vein HU did when he was still there.
If you like Green Day then I'd point you towards A Day to Remember. Don't know them that well, but they remind me of Green Day by falling into the (imo) somewhat rare category of good pop punk, though they also mix in a little metalcore influence which is nice. And you almost definitely already know them, but Paramore had some pretty great stuff too. Hoobastank and The Offspring are solid.
If you like Eminem try out Hopsin's "Ill Mind of Hopsin" series, specifically 5, 6, and 7. Rap with a message, good stuff.
Going into slightly heavier stuff, Slipknot are a must. Can be a bit hit-and-miss in my opinion, but when they're good, they're incredible. Check out Sulfur, Psychosocial, The Devil in I, and Duality - if you don't like those, maybe it's not the band for you, but if you do then explore them further. Five Finger Death Punch, Korn, Avenged Sevenfold, Alter Bridge, and In This Moment (female scream vocals done well, ooft) are all fucking great bands, you should listen to all of them.
And to finish off, to chuck in a few bands that hadn't crossed my mind until now and I don't wanna insert randomly: Bon Jovi for classic rock, Iron Maiden for classic metal, Tenacious D for sheer Jack Black awesomeness and hilarious lyrics combined with some surprisingly good music, and, of course, Nirvana. Because Nirvana were gods.
That should give you one hell of a lot of material to work with. ^^
EDIT: If you want a slightly truncated version with just my "best picks" from all the above:
Breaking Benjamin
Three Days Grace
Stone Sour
Evanescence
Halestorm
Slipknot
FFDP
Alter Bridge
In This Moment
Tenacious D
Nirvana