Act III is pretty hectic. The main town you'll be using is constantly under siege and enemies even pour into the town later on. Begin by talking to Tyrael.
Walk down the stairs to the Armory.
There will be a little scene between Adria and Leah playing out here. You can watch it or go into Bastion's Keep Stronghold.
Explore the city until you're familiar with it and then take the path Northeast and head into the Skycrown Battlements.
You'll have to light the five bonfires here. Follow the path and light the bonfires along the way. I've got all five bonfires lit in this picture and their locations. They should be exactly or at least relatively near what my map says. The orange fires are the bonfires.
Talk to Sergeant Dalen.
Walk into Stonefort.
Follow the path until you reach Captain Haile and talk to him. Enemies will spawn from the grate and then from the elavtor that comes up. Kill them and proceed to the first catapult you need to raise.
At the first catapult, talk to Sergeant Burroughs. Go Northwest a little and kill the demons attacking the barracks.
Once you kill the demons and free the workers, they'll start to raise the catapult themselves. Just stand beside them and protect them while they do their thing. The less that die, the faster it'll raise. Once you finish here, continue to the second catapult.
At the second catapult, talk to Lieutenant Lavail. He'll tell the the workers in the barracks to come out and start doing their jobs. Same as the first one, protect the workers, just there are more enemies this time.
After you finish with the second catapult, head onwards to the third catapult and talk to Lieutenant Clyfton.
There won't be any workers helping you with this catapult. It'll just be you, Lieutenant Clyfton, and tons of demons. You can click on the winch every few seconds, but don't forget to kill the demons that spawn as they'll continue to spawn until the catapult is raised.
After you raise the third and final catapult, teleport back to town and talk to Tyrael.
Head Northwest and kill the demons attacking the stronghold, then enter from where they came, The Keep Depths Level 1.
Explore this area until you find The Keep Depths Level 2 and enter it.
Explore this area until you find The Keep Depths Level 3 and enter it.
Explore this area until you find The Larder. This is your first main boss to fight in this Act. Ready yourself and enter.
He has two main abilities he uses. He summons little yellow gobs like the ones here and he puts clouds of poisonous gas on the ground at your location from time to time that last for a while. If you take too long to kill him, he'll increase his rate of spawning poisoning clouds and you will eventually not be able to move without running into a cloud.
Once you defeat him, teleport to town and talk to Lieutenant Lavail.
Head right and into the Armory.
A scene will play out. Talk to Leah when you can.
Head back to the stronghold and go Northeast, straight across the bridge and into The Arreat Gate.
Follow the path until you reach Sergeant Pale and talk to him.
Explore the area until you find the Bridge of Korsikk and cross it. Explore the next area and find and destroy the three Demonic Ballistae and the Trebuchet. Their locations are on the map shown here, or at least one possible set of locations if its randomized every game.
Head Northeast until you reach Rakkis Crossing. Tyrael will be here. Talk to him and then continue following the path.
Follow the path until you reach the Edge of the Abyss. This is another major boss fight. Prepare yourself and enter.
The Siegebreaker Assault Beast has few troublesome abilities he can use. One of his abilities is to grab (if you're close enough) and just squeeze you for pretty decent damage. He'll also summon these giant demons on the sides that will spawn the demon beserkers. If you can't kill the boss fast, I recommend killing them first so you don't get overwhelmed with enemies. He'll also do some ground sweeps with his claws in front of him, easy to avoid.
Once you defeat him, head a little south and talk to Adria.
Go to where the Siegebreaker began and enter the Arreat Crater Level 1.
Follow the path until you reach the Tower of the Damned Level 1 and enter it. Explore it until you find the Tower of the Damned Level 2 and enter that.
Explore the Tower of the Damned Level 2 until you find the Heart of the Damned and enter it.
Follow the path until you reach the Sin Heart here. Cydaea will drop down and defend it. Damage Cydaea until she runs away and the magic guarding the Sin Heart will dissapate. Kill the Sin Heart.
Go Northwest and follow the path until you reach the Arreat Crater Level 2 and enter it.
Follow the path until you reach the Tower of the Cursed Level 1 and enter it.
Follow the path until you reach the Tower of the Cursed Level 2 and enter it.
Follow the path until you reach the Heart of the Cursed. You will face off against Cydaea for serious this time and won't be able to change your abilities inside until you kill her. Prepare yourself and enter.
Cydaea is pretty easy. She'll shoot some bolts of venom at you sometimes, spawn some minions, but not much to worry about. Kill her, then kill the Sin Heart.
Go behind the destroyed Sin Heart and follow the path until you reach The Core of Arreat and enter it.
Follow the path until you reach the Heart of Sin. This is the final boss in this Act, the Lord of Sin.
Azmodan will have three abilities you will need to watch out for. His most common one is this fireball that chases you around a few seconds then explodes on either hitting you or in mid-air if it doesn't reach you. He'll summon demons and portals for demons. You should kill them both, demons and demonic portals, if you can't kill Azmodan fast enough. As you are fighting Azmodan, he will leak this red/black ooze heading towards the center of the map. If you can't kill Azmodan fast enough, the entire area will be covered in the ooze and you'll have no safe spot to be in.
Once you kill Azmodan, collect his Soul and teleport to town.
Talk to Sergeant Lavail.
Head to the Armory.
Enter the Bastion's Keep Watchtower.
After the cutscene, click the Portal to Heaven and you'll be done with Act III and onto Act IV!