Main Progression

NOTE: Much credit to u/Kartelant for posting this originally on reddit

To Undead Parish

 * Get to Darkroot Basin one of two ways:
 * From Undead Burg
 * Head up the stairs from Firelink and proceed through Burg as normal. Make sure to grab the Burg Latchkey by rolling through the barrels in the very first part of the burg and exploring around, as you'll need it to access a bonfire and save Griggs.
 * Defeat Taurus Demon, receive Watchtower Key.
 * Hellkite keeps too close a watch on the bridge these days, so use that key to go down to where Havel usually is and open the door to Darkroot Basin.
 * From New Londo
 * Explore New Londo for the key to Valley of the Drakes.
 * Run down the valley hugging the right wall and take the elevator up to Darkroot Basin.
 * Go up the cliffside paths from the basin towards Andre.
 * There is a new boss in the room just before Andre's usual room containing a Titanite Demon. Defeat it to proceed.
 * Welcome to the Parish! Sen's Fortress is already open to explore, and Andre is just inside its main gates, so don't miss him.
 * Gargoyles are missing right now, but you can fight them after you acquire a set of heavy armor (or at least just the gloves).

Through Darkroot Forest Sealed Door

 * The crest of Artorias is no longer in the game. Instead, the door only opens to "compatriots of the abyss walker".
 * Get any of the following items:
 * Hawk Ring - Drops from the giant that normally opens the doors to Sens Fortress
 * Hornet Ring - Just after pinwheel, nestled in a big sarcophagus next to the mimic bonfire.
 * Leo Ring - Drops from Ornstein
 * Silver Pendant - Drops from boss at Grave of Artorias
 * Wear one of the rings or have the Silver Pendant in your inventory and interact with the door to open it.

To New Londo

 * Take the lift down to New Londo.
 * New Londo is extremely easy right now, there are very few enemies. Explore to your heart's content, and optionally kill Ingward for the Key to the Seal if you'd like to hop down to the Four Kings early. But beware - draining the water of New Londo has some rather dire side effects on the previously peaceful area.

To Blighttown

 * Explore New Londo for the key to Valley of the Drakes.
 * Enter Blighttown through Valley of the Drakes, and proceed as normal.

To Depths/Lower Burg

 * Travel up backwards through Blighttown to reach the door connecting it to the Depths.
 * Examine the door three times to get a key and open it.
 * Gaping Dragon isn't here right now - explore the area and continue upwards to Lower Burg.
 * If you'd like to fight it, find the Peculiar Doll behind the group of torch hollows in Lower Burg and come back.
 * Capra's old boss room is a bonfire room now. Feel free to punch that fog wall.

To Anor Londo

 * You can explore Sen's Fortress as soon as you reach it, but you cannot go to Anor Londo yet.
 * There are two ways to get to Anor Londo:
 * Get the Pale Eye Orb from Patches's first encounter or the pit he kicks you down, then return it to its resting place in a coffin in the Catacombs, to return the pale servants to their post and enter via Sen's Fortress.
 * Go to Painting of Ariamis by exploring lower burg and depths until you get a painted doll, then fighting Gaping Dragon. Upon exiting the painting, you are in Anor Londo.
 * Note that you can explore Duke's Archives immediately upon entering Anor Londo, as it is not blocked by sunlight fog.

Obtaining the Four Lord Souls
This is pretty much the same as usual. Kill the four bosses, put their souls in the Lord Vessel, fight Gwyn. Except now it is possible to kill most of the Lord Souls bosses without beating O&S or ringing both bells.

Lost Izalith and the Lava

 * After killing Quelaag, proceed to kill Ceaseless and go through Demon Ruins as normal. Demon Firesage and Centipede are not in this area.
 * Upon activating the shortcut lift back to Chaos Servant covenant, go and check the shortcut to Lost Izalith, it should be openable.
 * Explore Lost Izalith as you please.
 * To get in the lava and kill Centipede demon, chug a Sunlight Elixir (it gives super fire resistance) and run to where the illusory wall that usually has a bonfire behind it is located. A daughter of chaos sits wounded within, and casts a spell on you to allow you to fight in lava just fine, so you can kill Centipede and get the orange charred ring to explore the rest of the lava area.