Cave Story/Labyrinth

You wake up in another cave. There will be some blue robots nearby who you can talk to, but they don't have anything interesting to say, so shoot out the breakable block and save. Once again, this part is going to be difficult if you don't have the level three Machine Gun, but the Spur is worth it. Equip your newly acquired Blade; it can kill the purple critters in one shot. In fact, it can kill almost anything in the labyrinth in one shot. You will be relying on it a lot, so don't neglect it (note, though, that when it goes from level 1 to level 2 it decreases in range. However, it becomes more powerful and gets plowing abilities, so that's good).

Jump up the platforms, and try to kill the critters before you aggravate them, because dodging their projectile attacks while not falling is a major pain. About halfway up, there will be an area on the left with three critters. One of them is standing in front of a life capsule, so take them out and get the capsule. Near the top, you will run out of blocks, but then a big block detaches itself from the wall and moves towards you. Jump on it to trigger another. Jump to the very top on these blocks to find a terminal, which opens the door. Jump down (don't worry about fall damage) and see that the blue robots (except the one which looks dead or out of fuel or whatever you'd call it) have gone through the door. Follow their lead, but don't forget to save first.

Reunions
You emerge from the doorway to see the scrapped remains of one of the blue robots. There will also be cockroach monsters, but they're pretty easy to avoid and die in one shot from the Blade. Watch out for the flying ones though; they have projectile attacks. You will find a save room soon enough with a bunch of non-aggressive cockroach monsters with whom you can speak. Talk to the one behind the counter and he'll give you the turbocharger if you have the Machine Gun, which is virtually mandatory, being that the Machine Gun recharges so slowly normally. If you don't have the Machine Gun, he'll offer to combine your Polar Star and Fireball for the Snake. The fact that this gun goes through walls is very useful in the labyrinth, but if you want the Spur, don't do it.

Keep going until you get to the next save room, and there will be a (kind of weird looking) physician and nurse, as well as Curley, looking somewhat distraught, on one of the beds. Talk to the physician to restore your health and get a key, and to Curley for more information about the Labyrinth, then head up to the door with a red cross above it. Be prepared - you're going to get into a boss fight in a moment. Open the chest for the cure-all, and Pooh Black will appear. After he's done, go back to the physician and give him the cure-all for Curley. Note that there is a high up passage (visible on the map) that will lead you to a secret door, which will give you the Arms Barrier, a very useful item if you don't want to hold out for the Spur, and useful anyway if you are. There are several ways to get to the door:
 * If you have the Machine Gun on level 3, boost upwards by firing it down.
 * Go ahead past Monster X, talk to Professor Booster, and get the Booster 0.8. This will cause you to lose your chance to get the Booster 2.0 later, and also keep you from getting the best ending, but it will allow you to use the Booster 0.8 to reach the door to the room with the Arms Barrier.
 * Hold out for the Booster 2.0, and come back to this area much later in the game.
 * Kill all the Gaudi in the room, then jump onto the bouncing purple Critter and try to land on it at the highest point of its leap. You'll take damage, but you'll bounce off the Critter (you may have to press Jump when you hit it) and you can land on the ledge which leads to the secret passage.

Next up, another boss fight! Monster X is an annoying one, but it's doable. The Blade and Missile Launcher work best here.

After he's done, go through the door and you'll hear the sound of a teleportation, and Booster will fall down the nearby ravine. If you follow him and talk to him (and you have no choice but to talk to him if you fall unless you have the level three Machine Gun), he will give you the Booster 0.8, then die. If he dies now, he won't give you the booster 2.0 later, so it's worth it to not let him die. You also can't get the third and best ending without the 2.0. If you don't want that to happen, save, and make the tricky jump over the ravine. You can always reset if you miss, but the trick is to run from the door and hit Jump as soon as you cross the red mark on the platform you're running across. Obviously, this is much easier with the level three Machine Gun.

In the next room, you will see Curley in front of a giant boulder, which is blocking the exit. To advance, talk to her, and then attempt to move the boulder with her. After your failure to do so, Balrog shows up. Curley seems to think Balrog will just help them with the boulder, and he almost does - but then he attacks. After you beat him, he says that he'll do it now, so you can move on. When he jumps through the ceiling, as usual, a save point and a recharge station show up, so use both and then go through the door.

The Core
Curley will actually be fighting with you this time. Note that she still has her Machine Gun, unless you swapped, in which case she has the Polar Star. Either way, it's level three, although she doesn't use the Machine Gun jetpack ability. She doesn't take hits, and she has an air tank, so she can't die, and can go through enemies. And if she has the Machine Gun, your enemies will be gone before you know it. This area will be crawling with bugs, and they can overwhelm you with numbers alone. The level three Blade is sublime in this area, but barring that, use the Blade anyway, or the level three Bubbler.

Soon enough, you'll get to an area with water everywhere and a door like the hermit gunsmith's door, which you can't find anywhere other than these two places. You can't go through the door though, so save and follow Curley. Use the terminal to open the shutters, then when #4 acts up, shoot it with your Polar Star/Machine Gun/Snake until it gets unstuck. Hit the first terminal again to lower the lift, and go through the passage for the next terminal to open the main shutter. Save at this point, because this next boss is not a pushover, especially if you don't have the level three Machine Gun or the Booster 0.8 (it's still worth it for the Spur and booster 2.0). Do not talk to Curley, unless you never want to see her again after this fight. Instead, go to the lower right corner of the field, and there will be something shiny there. Take it and find that it is a tow rope. This may not seem significant now, but it is. Now you can talk to Curley and initiate the boss fight.

Don't forget that the water rises and falls, so if you don't have the Booster or level three Machine Gun, it's going to be a pain to not drown, but keep at it. After the fight, Misery and the doctor show up and teleport the core away, and the water rises to maximum, and stays there. Curley is okay, she has an air tank, but you're drowning. The shutter is closed, so you have no choice but to let your air drop to zero. At this point, if you picked up the tow rope and didn't let Booster die, you receive a message about a plot within the plot, and your first hint about getting the third and best ending. Then you wake up to find that Curley has given you her air tank, and is now lying unconscious on the floor. If you have the tow rope, attempt to talk to her, and after getting no response, use the tow rope to get her out. If you don't have the rope, well, she's a goner. Either way, go back to the area with the tiger mouth door, and find that it is now open. Go on through.