The wooden door, as it turns out, is barred from the far side and thus inaccessible to you. The path down, though, is traversible, so head cautiously down the stairs- the landing at the bottom opens into a vampire dining room, where another fledgling waits- though thankfully alone. Slay her and you can rummage through the room for various potions, foods, ingredients, and a couple of leather strips. Curiously, some of the ingredients come up with a 'steal' message instead of a 'take' message, though there's no explanation on hand for why that is.
Continue down the semi-frozen stone tunnel here and you'll find that you come out into a two-level cave on the lower level. There is a vein of ore near the bottom of the ramp up, but mining is loud enough that you could be in danger- thankfully, no enemies are nearby, so you can gather some iron ore here if you feel like.
Ascend the ramp to the upper platform and you'll find that you're being ushered into a huge cavern, dripping with rime and lined with wooden platforms and railings. There are a number of enemies here as well, and this is one of those locations where stealth and sniping, be it with arrow or spell, truly shines.
The obvious target is the thrall that patrols up onto the platform directly across from the opening you are using to reach the cavern, but do be wary- there is at least one vampire on the cavern floor waiting to make life difficult for you, and several of them can summon skeleton companions. The vampire fledglings are by far the easier targets to take out, but will bring the guard down on your head.
If you're feeling especially brave, you can likely charge the fledglings and take them out before their guard reaches you, provided their summoned skeleton isn't too much of an impediment to you. The guard, of course, is very dangerous in melee, so you may find that this still leaves you weakened enough to make him very threatening.
Once you've cleared out the enemies and looted any corpses on the ground level, grab some supplies from the nearby barrels and make note of the louder howling nearby.