Yeah, go into the castle then make a hole and go left through it.
Once you hit the grassy patch that you can see the starter village from start digging down. Lots of gold and iron down there. --- XLB: CodyBlues. FC 8054-4057-7024-3905 Animal crossing FC 0603-5159-4579 First person to get it- Azrahai
There are indeed three dungeons in the Tutorial seed (in both the unpatched, and patched versions of beta 1.6.6) that I know of. Two of them have Zombie monster spawners, and the other is a Skeleton spawner.
Since I don't have pictures at the moment, you'll have to make due with the raw coordinates (thankfully, they're pretty easy to find by "digging straight down"... though you can just find the second Zombie dungeon by going in from the open side of the mountain across from the village).
The first, and easiest to find, is the Zombie dungeon hidden almost right under the world's default spawn point. Go to coordinates X: -15, Z: -60 and dig straight down to layer (Y) 17 . Be advised, however, that you WILL fall into part of the natural cave system directly below the spawn point... just keep digging until you hit the dungeon.
The next, and this is the one that Jin is talking about near the castle, is found by going to X: -97, Z: -127 and digging down to layer 42. This is the Skeleton spawner.
The last, and the one that I mentioned before, being "across from the village" is found at X: -201, Z: 38 , at layer 41. You can find it by either trying to dig straight down (I haven't tried this method, personally... because you can get to it even easier by...) or by entering the mountain from the side facing the village. There's a gap in the mountain on that side, where the frozen ocean stops right at the mountain... it's hard to spot, but it's there. You just follow the cave system going ~right~ from the entrance. It's VERY easy to miss the path, because you have to keep going along the right branches (and you risk falling pretty far down in a few spots)... just follow your coordinates on your map, bring a few torches, and when you get into the hollowed out area, look ~up~ at the ceiling... there will be a telltale sign, in the form of cobblestone. It's up ~in~ the "ceiling" near the coordinates I gave you, if you were to dig right down through the top.
And while it wasn't asked, there is also a Slime chunk on the Tutorial map that I found, around X:65, Y: 7, Z: 332. I dug a lovely 60x60x12 pit, that goes from X: 108 Z: 332 to X:48 Z: 332, and down to Z: 272. That thing is ~huge~. I've already ganked over a hundred slime (and got 112 Slime Balls for my efforts)... but I have failed to isolate the ~exact~ spot the Slimes appear in... sometimes they appear ~exactly~ at X: 65 Z: 332... but other times they appear anywhere between X: 48 ~ 108... ~very~ annoying.