You basically need two things to get the Best Ending. Massive spoilers.
You need the Booster 2.0 and you need Curly. All the big points you need to make sure to not screw up are (not) in bold below.

Booster 2.0
 -  For Professor Booster to survive, he needs to keep the Booster 0.8.
    So you can't take it from him - this means when you see him fall past you in the labyrinth, don't drop down.
    Instead, jump across the gap (you can do it with a good moving start or just use Machine Gun lv. 3 if you have it) and continue onwards.
 -  You get the Booster later on.

Curly
 -  When you access the Labyrinth Core room for the first time, do not immediately investigate the robots (which triggers the boss fight).
    Instead, drop down to the bottom right corner of the room, there should be something shiny.
    Pick up that Tow Rope, you'll need it.
 -  When you defeat the Labyrinth Core boss fight, Misery floods the room and Curly gives you her oxygen tank.
    If you picked up the Tow Rope I had just told you to, go examine Curly and you should use the item to grab Curly and bring her with you.
 -  In the ensuing water ways sequence, make sure to stop at the cabin on the way.
    Put Curly on the bed and examine the bookshelf to find instructions on how to drain the water out of her.
    Go do just that, and remember to take her with you when you leave. She'll drift away after fighting Ironhead, this is expected.



~Later~
 -  Much later on in the Plantation area, head to the bottom right of the whole map and you'll find Curly and another guy.
    Talk to both of them to get caught up on the story and learn about a mushroom you need to grab in the Mimiga Graveyard.
 -  Backtrack to the Mimiga Graveyard and use your fancy new Booster 2.0 to get to the previously inaccessible door. There, you'll find a mushroom.
    Pass his quizzical bombardment, and you'll get a Mushroom Badge.
    Examine that badge and then talk to that conniving mushroom so that you can get the real mushroom. (By force.)
 -  Return to the Plantation and give Curly the mushroom. After you finish your conversation, talk to her again to get the Iron Bond.
    Do NOT forget to get the Iron Bond. This is required.