-Easiest Heroic: in general, you're going to want to start in the Slave Pens. I've heard good things about Underbog and Old Hillsbrad (Ramparts is comparably easy as well, especially if you've got your FR gear on), but really I've found that Slave Pens is a good indicator of whether your group is ready for Heroics or not, and it seems like Cenarion rep is pretty easy to come by, so you've got lots of keyed folks to choose from.
-Best place to get badges: Heroic Mechanar. Even the Gatekeeper minibosses, who don't give good loot in the normal instance, will drop Badges for you in Heroic, which means that if you skate through this place in under an hour, you're earning five badges in that time.
-In Heroics, it's not the bosses you have to worry about, it's the trash. Trash hits hard (really hard), so there's no room for error (read: letting the mobs go after clothies). If you don't have the standard CC you need (and you should-- heroic groups are best when every class type is well-represented), then fear is your friend, because in most heroics two hits are enough to cause a wipe.
-So how do you keep wipes from happening? The usual-- use the raid icons, plan things out, and don't pull aggro away from the tank. Establish a kill order and follow it. Have your mage or hunter pull their sheep or trap away from the rest of the group (with a rank 1 frostbolt or a distracting shot) so no one gets confused. It's basic stuff, but heroic enforces it to the max: no one should be getting hit but the tank, and that way the healer only has to heal one person.
-If you don't have the gear, come back when you do. Heroics are merciless, and that's why they're so damn fun. But if you don't have what you need to finish off an instance, that fun melts away, wipe after wipe. If your group isn't ready to start heroics, there's an easy way to fix it: go run Shadow Labs or Arcatraz (or Karazhan, if you can) again, and then come back with better gear.
I love running Heroics-- they were a terrific idea by Blizzard to put a new challenge into what are quickly becoming familiar instances (here's hoping, as always, that we get some more Heroics in the even older instances). The truth is that as gear continues to get better, Heroics will get easier and easier, so it could be that the fun to be found in there is right now-- when people are first starting to learn the tricks. After a while, Heroics may become old hat. But until then, players seem to be having a great time meeting the challenge and earning those Badges.