Cave Story/The Mimiga Village

Yamashita Farm
After viewing a short dialog between Toroko and King, go to the Yamashita Farm by walking right from where you fell going up whenever you can. You will eventually find the door to it.

Once in the farm, talk to Sandaime if you want, then walk to the end of the farm to find another heart container. After you get it, walk out of the farm and continue onwards to the reservoir.

To get to the reservoir, walk back from the farm entrance to the location where you fell. The door to the reservoir is directly to the left of it and a little below. Go into the reservoir

The Reservoir
From the door, walk along the path to your left. While walking, you have two options you can take. First, you can take the dry route and jump along the platforms (and talk to Kanpachi). Otherwise you can run along the bottom of the reservoir, it honestly doesn't matter.

At the bottom left of the reservoir you will notice a sparkling section of water. Press while in front of it to pick up the Silver Locket. Proceed back the way you came, and you will see Toroko run out the door after talking to you. Follow her back into the village.

Shack
After viewing another dialog between Toroko and King, go to the building in the lower right hand corner of the village. Go inside and walk forward a bit. Toroko will jump out and attack you, but a shot or two will defeat her. Talk to her afterwards to trigger the next event.

Balrog and Misery will come barging through the door and Misery will kidnap Toroko. Then you will have the option to fight Balrog if you want (strategy here). The reward for beating him is 5 exp for your polar star. After you are done, walk out of the shack.

The Map System & Arthur's Key
From the shack entrance walk left until you reach a long vertical shaft that you can climb to reach the save point in the middle of the town. From there, walk right a little bit until you reach a gap in two platforms that you need to jump. Fall down that gap and land on the suspended platform below it. Jump on the series of hanging platforms to reach a treasure chest containing the Map System.

Fall down and you will end up at the shack again. Repeat what you just did and go all the way back up to the gap where you fell down. This time however, jump over the gap and proceed to Jack, who is guarding the door. Talk to him and he'll run away. You can either follow him into the Assembly Hall for some backstory, or proceed right along into the Graveyard.

Once in the Graveyard, follow the path and defeat the enemies along the way, although only the Grave Keeper will give you experience points. While traversing the Graveyard, you may notice a door high above that you cannot reach. You can't get there yet, so just keep heading left along the platforms. Jump onto the platform that contains Arthur's grave and again press on the sparkling tile to collect Arthur's Key.

If you re-enter the graveyard after leaving, the Grave Keeper (and other enemies) will respawn, effectively allowing you to farm experience points by killing the Grave Keeper many times over.

Arthur's House
Arthur's house is located in the very bottom left hand corner of the village. When you reach the door, press on it to open it with Arthur's Key. It is recommended that you heal and then save in his house. Walk over to the lit computer screen and read it to be asked if you want to open access to Egg Corridor. Choose yes.

You can also walk down and examine the bottom right corner of his house to get the message: "...Flowers?". When you are ready, walk up to the teleporter and choose to warp to the Egg Corridor to be teleported there.