Gathering the quests:
Horde & Alliance:
NOTE (Important): The quests marked in Orange are the ones you must have up (or already completed) before you enter the instance for the first time. Try doing the Disharmony of Flame (Horde) / Overmaster Pyron (Alliance) quest as soon as possible, so you can get access to all the Orange quests listed. If there is still a quest you can't access, don't worry, come back later after your first BRD run and check again, it should get unlocked at some point.
Dark Iron Legacy (Elite) , given by Franclorn Forgewright, outside the instance (see map). Franclorn is a ghost, so in order to talk to him, you must be in ghost form (die pls!). It's important that you get the quest from him before entering the instance, because it's the only way to get the Shadowforge key, which unlocks the doors to the interesting areas of BRD. Don't bother with any quest not marked in orange for now, and follow this guide blindly.
Ribbly Screwspigot (Elite) , given by Yuka Screwspigot in Flame Crest, Burning Steppes.Pre-req for this quest is to complete the quest Yuka Screwspigot, given by Yorba Screwspigot in Tanaris.
The Love Potion (Elite), given by Mistress Nagmara inside BRD, at the Grim Guzzler Tavern. This quest requires you to gather different components in different places of Azeroth. Completing it is worth it, besides the item reward, Mistress Nagmara has another function. When you have completed the Love Potion quest, on your next runs to BRD, she will open the backdoor of the Tavern for you if you talk to her. That way you won't need to kill Phalanx everytime you go through the Grim Guzzler.
The Heart of the Mountain (Elite) , given by Maxwort Uberglint, at the Flame Crest, Burning Steppes. This quest's objective is to open every coffer in the BRD Vault. For that you need your group to have gathered 12 Vault Coffer Keys. They can drop from any Iron Dwarf in BRD.
A Taste of Flame (Elite) , given by Cyrus Therepentous. Objective is to kill Bael'Gar in BRD (see map) and to capture his fiery essence with the help of an Altered Black Dragonflight Molt. How to get the Dragonflight Molt:
A long serie of quests starting with Kalaran Windblade, in Searing Gorge, you can find him North of the area called 'The Cauldron' (loc 39,39). He will have you do a bunch of quests, starting with  The Flawless Flame, until you get to the point where you get the Torth of Retribution, and then he will send you to Squire Maltrake. Squire Maltrake will give you the quest Set Them Ablaze!, you will have to use the Torch on the braziers in each of the 4 towers. After you finish the quest, Kalaran will turn into a dragon (not an aggressive one, don't worry) and talk to you, and a chest will appear on the ground. Opening it gives you a quest, complete it and you will get a box. Inside this box is the Black Dragonflight Molt. Now that you have the Black Dragonflight Molt, go talk to Cyrus Therepentous, he is in the Eastern part of Burning Steppes, in a cave (loc 39,39).
Disharmony of Flame (Elite), given by Thunderheart, in Kargath Outpost, Badlands. Objective of this quest is to kill Overmaster Pyron, who can be found outside of the BRD instance, he roams around the bridge leading you to the instance portal.
Disharmony of Fire (Elite), given by Thunderheart as well. This quest is the follow up of Disharmony of Flame.
Objective is to kill Lord Incendius (details about this can be found in the walkthrough) and to loot a tablet from him.
KILL ON SIGHT: Dark Iron Dwarves (Elite) , given by the WANTED poster in Kargath, Badlands. Anvilrage Footmen, Warden and Guardsmen can be found in the first areas of BRD, you can hardly miss them. This quest leads to the following:
KILL ON SIGHT: High Ranking Dark Iron Officials (Elite), given by the same WANTED poster after you completed the previous quest. You can find those Dark Iron Officials mostly on the Dark Iron Highroad (close to Bael'Gar, check map).
When you have both KILL ON SIGHT quests done, a new quest will unlock:
Grark Lorkrub (Elite), given by Lexlort in Kargath, Badlands. This quest will send you to Burning Steppes, you can find Grark Lorkrub inside the Blackrock Orc fortress, east of blackrock moutain. Finishing this quest unlocks a last one:
Operation: Death to Angerforge (Elite), given by Warlord Goretooth in Kargath, Badlands. (See above for pre-requisites). Will describe later on how to get to General Angerforge and how to kill him.
Commander Gor'Shak (Elite) , given by Galamav the Marksman in Kargath, Badlands (he is on top of the Guard Tower). If you don't get this quest, then try doing Disharmony of Flame to unlock it. After you complete the Commander Gor'Shak script sucessfully (Commander survives), go talk to Kharan Mighthammer. He is in the opposite jail. Listen to his story and then you will be sent to Thrall.
The Royal Rescue (Elite), given by Thrall in Orgrimmar, the pre-req for this quest is to complete the Commander Gor'Shak quest. The objective is to Kill Emperor Dagran Thaurissan and to rescue Princess Moira Bronzebeard. (Will be described later)
Lost Thunderbrew Reciepe (Elite) , given by Vivian Lagrave in Kargath, Badlands. Will describe how to get that reciepe later in the guide.
The Last Element (Elite), given by Shadowmage Vivian Lagrave in Kargath, Badlands. Pre-req for this quest is to complete Disharmony of Flame. Objective is to gather 10x Essence of the Elements, which can be dropped by Golems and Elementals inside BRD.
The Rise of the Machines (Elite),given by Lotwil Veriatus, East of Kargath, Badlands. Pre-requisite for that quest is to complete The Rise of the Machines, given by Hierophant Theodora Mulvadania in Kargath, Badlands. The pre-req quest can be done solo at lvl 54+.
Overmaster Pyron (Elite), given by Jalinda Sprig, at the Alliance outpost in Burning Steppes. Overmaster Pyron is a Fire Elemental, you can find him right before the intance portal to BRD.
Incendius! (Elite), given by Jalinda Sprig as well. This quest is the follow up of Overmaster Pyron.
Hurley Blackbreath (Elite), given by Ragnar Thunderbrew at the Tavern in Kharanos, Dun Morogh.
The Good Stuff (Elite), given by Oralius, at the Alliance outpost, Burning Steppes. Any humanoid type inside BRD can drop Dark Iron Fanny pack. You will complete this quest on your way through the instance.
Marshal Windsor (Elite) , given by Marshal Maxwell, at the Alliance Outpost, Burning Steppes. Objective is to find Marshal Windsor in BRD. (see map)
Abandoned Hope (Elite), given by Marshal Windsor in BRD as a follow-up from previous quest.
A crumpled up note (Elite), found on any mob in BRD. Right-Clicking the note starts the quest, which requires you to go and talk to Marshal Windsor again. (It's rumored that the note starts to drop only after you finished the two previous quests.)
A shred of Hope (Elite), given by Marshal Windsor after bringing the Crumpled up note to him. Objectives of that quest is to gather the marshal's lost information. For that you have to kill General Anvilrage and Golem Lord Argelmach (details about this in the walkthrough).
Kharan Mighthammer (Elite), given by King Magni Bronzebeard himself, in Ironforge. Kharan Mighthammer is in one of the jails in BRD. (see map) The pre-req for this quest is to complete The Smoldering Ruins of Thaurissan given by Royal Historian Archesonus, in Ironforge. That quest is easy to complete solo, go to the Ruins of Thaurissan in the Burning Steppes and look around for little stones and click them.
After finishing the previous quest (talking to Kharan Mighthammer), he will give you the quest The Bearer of bad news ,go back to King Magni Bronzebeard who will give you the follow-up quest:
The Fate of the Kingdom (Elite), objectives are to kill Emperor Thaurissan while letting Princess Moira Bronzebeard alive (even though she will be beating on your group). Details about this later in the walkthrough.