Grab the Dungeon Cell Key from the corpse in front of you and use it to escape your cell. Make sure to read all the orange messages as you continue on, for they are left by the game itself and serve as mini tutorials. Don’t worry about the undead at the moment, for they won’t attack unless attacked.
Proceed (you can’t get lost) until you’re outside. Light the nearby bonfire, then rest at it. Resting at a bonfire will always heal you, replenish your flasks, refill you spells, and respawn all minor enemies. You can also level up and do other things, though not at the moment. Enter the large wooden double doors and be ready for an ambush.
Run past the Asylum Demon and through the small doorway along the left wall, which will close behind you. Or, just be a badass and Chuck Norris the monster right now…
OPTIONAL BOSS: ASYLUM DEMON
Above: Defeating the Asylum Demon during your first encounter (fists and fire methods)
You can defeat the Asylum Demon here and now, without a decent weapon or Estus Flasks, for a bonus reward: the Demon’s Great Hammer. Switch to your fists (which swing faster than the sword hilt and cost less stamina), lock on to him, and circle strafe around him to his left. If you stay close and behind him, you’ll be safe. Two of his four attacks only hit in front, so if you keep at his rear, they’ll whiff. When he raises his hammer to swipe it horizontally to his left, roll towards his right side in order to stay behind him. When he jumps into the air, back away to avoid the ensuing slam attack. Attack when you can and you will eventually defeat him. Just make sure not to get backed into the corner like we did.
Also, if you chose Black Firebomb instead of Master Key as you gift, you can make quick work of the demon. In the above video we show the fists method then, for the sake of time, the Firebomb method. (Disclaimer: You won’t have the stats to wield the Demon Hammer weapon for a looong time, so don't worry if you don't feel up to this. You can always get the hammer in New Game +, when you'll be better equipped.)
Just after reaching the second bonfire, you’ll find yourself at the start of a long corridor. Aside from the Asylum Demon you just ran into, this is the first time you can be hurt by an enemy: be wary of the archer at the end of the corridor. Quickly run forward and into the room to your left. Grab your shield from the corpse, put it in your left hand using the equipment menu, and then press left on the d-pad once you’ve exited the menu to equip it.
You can now block with RB. Hold this button down and advance on the archer. He’ll turn tail and run; you can sprint after him and take him down quickly, or catch up with him later. Don’t forget to check the corpse on the ground near the archer’s initial position for your class’s primary weapon (a Hand Axe for Pyromancers). Continue on and traverse the white light (which we’ll refer to as “fog doors” from now on).
Head right and move slowly up the stairs. An undead will push a boulder down at you once you pass a certain height, so be ready to roll off (tap B) to the right. The boulder will have busted through the wall at the bottom of the stairs. Enter this room and speak to the wounded knight. Hear him out. He’ll give you precious Estus Flasks (your main source of healing) and the Undead Asylum F2 Key.
Ascend the stairs again. There will be a lone undead wielding a sword. Kill him and pass through the gate. Grab the Pyromancy Flame from the corpse on your right and equip it in the secondary slot of either hand (we prefer the right). You can now cast Fireball 8 times before resting at a bonfire, however, save it for the boss.
Around the corner to the left there will three undead: two with swords and one with a bow. Catch their attention and retreat so you can defeat the melee fighters without worrying about the archer, then return to easily dispatch the bow-wielder.
Now that you’re better-equipped, you should enter the fog door to fight the Asylum Demon. You can proceed past the fog and take on an undead soldiers, but unless you want the souls and parrying practice, there’s no point to defeating him as the door he guards cannot yet be accessed.
BOSS: ASYLUM DEMON
With your weapon and Pyromancy Flame, this battle is extremely easy. You can adhere to the above strategy from the “optional” section, but you really need only get close, wait for any attack, roll away, then throw a fireball while the demon is in his recovery state. Don’t forget to equip your axe and press RB just after falling from the initial ledge to do a plunging attack. This puts you at a huge advantage right out of the gate. Killing the demon garners you one Humanity, 2000 souls, and the Big Pilgrim’s Key. Use the key on the large blue door nearby and run forward to the edge of the cliff.