Most players by now that started rolling to 70 when the expansion hit are probably either just starting or in the middle of running their first few Heroic dungeons. I fall into that group-- my guild has had a few folks running Heroics for a while, but the majority of us are just now starting to pick up the Revered rep needed in the different factions to make running Heroics a regular possibility. So here's a few tips I've picked up just in the past few weeks or so about how to get started running the most challenging 5mans in the game. Please feel free to add your own below.
-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.