Existing achievements (e.g., Explore Silverpine Forest) are being changed to reflect the new sub-zones and quest hubs being added to the game, so bank on seeing a lot of new requirements. Blizzard is changing the last batch of zones as we speak. Interestingly, the new Cataclysm zones that are geographically part of existing continents (e.g., Mount Hyjal) are not part of the old world exploration achievements; they've got their own separate achievement, below.
Explore Cataclysm Again, pretty routine. New zones -- go explore! It's been yoked to the World Explorer meta-achievement, though, so new players will have more ground to cover than someone who has the achievement right now.
Eastern Kingdoms Dungeon Explorer and Kalimdor Dungeon Explorer Blizzard's toyed with the idea of forcing characters to "discover" dungeon entrances before being able to queue for them in the dungeon finder, and that's currently necessary for all 80-85 characters on the beta. You'll note that the Deadmines, Stormwind Stockades, Ragefire Chasm and Wailing Caverns aren't on these lists. Blizzard kept them off, I assume, because it was always a tall order to ask characters at the appropriate level ranges to travel across the world (and often across much higher-level areas) just to queue for a dungeon. In the case of Stormwind Stockades and Ragefire Chasm, it's also next to impossible for low-level characters to get to dungeon entrances in enemy cities. However, in the latest beta patch, Daelo (lead encounter designer) has stated that Blizzard's removed the dungeon discovery requirements for all pre-Cataclysm dungeons, after players pointed out the number of problems it caused. The achievement may stick around as an extra bonus.
Outland Dungeon Explorer The Outland version of dungeon discovery achievements. As an aside, if you don't remember or know which dungeons you're eligible to queue for, pop open your dungeon finder, choose the specific dungeons option, and you'll see whatever's in your level range pop up. Anything that hasn't been discovered yet will have an X next to it, and you won't get it. On the beta, this has actually been pretty useful in that you can deliberately limit your exposure to dungeons you know you're not geared to do.
Northrend Dungeon Explorer As above.
From Hell's Heart I Stab at Thee Someone's been reading some Moby Dick lately, I guess. The deal with this achievement is that there are a few enormous Whale Shark mobs making their serene way around Vashj'ir. Yes, there are more than one; they're basically the Fel Reaver/Storm Giants of Cataclysm. In a reversal from their apparent behavior earlier in the beta, they're not actually hostile to players despite appearing to be so (you can swim right through them), and they are technically even soloable if you're a class that can kite.
Surveying the Damage This is the achievement we initially thought would cover all of old Azeroth, but it's more accurate to say that most of old Azeroth has changed without necessarily being damaged in the meantime. That's not the case with any of the spots listed here; most of these places have been blown to hell (the Cenarion Wildlands in Desolace are a notable exception). The scale of destruction is also pretty unreal, and this is the sort of achievement best left to characters already rocking a flying mount.
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