When you get to Halloween Town there will be a cut-scene. Nevermind the Heartless here, they won't hurt you. Whem you regain control, go north then west into the small gate. Get the chest here for puppies 69-71. Climb the stairs and pull the rope for the doorbell 3x to get an Elixir.
Now proceed to the lab entry door and go up the stairs. The professor will say he needs some memory and tells you to go find Sally. When you regain control, check the bookshelf to get a Torn Page. Jack will also join you here.
Exit the place and there'll be a cutscene between Jack and the mayor saying that something went wrong, so now you'll have to get rid of those previously harmless Heartless. Proceed to the Graveyard. There will be a swarm of enemies for you to kill. When they're all dead there will be a cut-scene with Sally. She'll give the forget-me-not item which is the memory. Take it back to the lab. There will be a cut-scene.
The professor will ask you to find one last ingredient, the "surprise". There will be a cut-scene with Oogie Boogie. Go back to the graveyard and jump in the coffin. Now you have to check the tombstones. Look at the order in which the ghosts rise and examine the tombstones in that order. The big pumpkin will open up and the treasure chest will contain a jack-in-the-box. Take it back to the professor.
After the cut-scene you'll need to find those kids. Go to the graveyard again and there'll be another cutscene. Go to where the open pumpkin was and examine the door. Inside explore the new area and you will find that there is a platform to be lit. Light it with your fire magic. Next go up the long pole and examine the structure that is sticking out of it. You'll be taken to a new area so go to the far left and light the platform. Cross the bridge and there will be a long cut-scene.
In this new area walk up the bridge and examine the door. Once inside there will be a red trinity mark and a treasure chest containg a ether. Light the lantern in that room. Jump on the platform. Your only way here is up, so keep going until you find the Playroom.
Here you'll find Lock, Shock and Barrel and they'll fight you. There isn't much strategy in this battle so just beat them up. Try talking to one of them and they'll accidentally tell you about the lever. So examine the machine on the side to find it. There's also a save point here. Save before leaving.
It's time to find the "green door". Just fall off the ledge and try landing on the grayish-black path to find the Torture Chamber. Go in.
You'll see a cutscene where Boogie swallows the heart and attempts to summon some Heartless. You'll fight him after that.