Cave Story/Sand Zone
Head left in the Sand Zone. You'll see someone spot you and run off--who could it be?
To find out, continue left and then use the platforms to head up, being careful of the enemies along the way. They like to hide off-screen or behind grates, but will only move if you enter their line of sight, so watch out. At the top, break through the blocks and go left some more.
Enter the building to start a short cutscene, then prepare to fight! Look here for a strategy. Afterwards, Curly will realize you don't want to hurt anyone--provided you tell her you aren't there to hurt anyone, of course. She'll offer you her Machine Gun in exchange for your Polar Star--if you are new to the game, you should probably take it.
Talk with the mimigas, then enter the room and grab the dog by the bed. Save and heal if you need, then head out to the rest of the Sand Zone.
 Lift the Spell on the Sunstones
Go right, destroying the blocks to progress, until you reach the area with columns of breakable blocks. Shoot a path into the first open area and either dodge or kill the Polish inside. Be careful of the sandy pit--any time you see a pile of bones on sand, it means there is a sandcroc waiting for you, and they do a lot a damage per hit. In the second column of breakable blocks, a heart container lies at the top--grab it and move on.
At the last column, make another staircase and climb to the top. Shoot out the blocks on the platform to the right, then jump over to it. This is easiest if you leave a level below the unbreakable blocks to run and jump from. Next, hop up towards the paw print block on the wall--you'll go right through! Behind the stalactites you'll see a floppy disk; if you need a few hearts, shoot to the right of the floppy disk. Save if you'd like, then drop down to the next area.
Now you are at the breakable Xs. Shoot out the spinning suns but destroy as few blocks as possible so that you don't have to fall into pits with a sandcroc or go chasing experience. This part is actually easier than the columns. At the end, be careful of what you shoot--take your time and destroy the blocks that will let you jump to the floating platform so you can heal. (See the screenshots for help.) The second platform has a sandcroc in it that is almost impossible to avoid, so don't jump in it at all. At the large wall of breakable blocks, be careful: there are enemies in them that will fall on you if you aren't careful.
Hopefully your weapons and health are full or near full, because you are about to fight a boss. After a short cutscene where Misery almost battles you, Omega appears, making it look as though maybe she doesn't have to fight you after all. After the battle a notice appears informing you that the spell on the sunstones has been lifted. Go back towards the beginning to see what it means.
 Finding Jenka (and her puppies!)
After you have defeated Omega, on your way back to the beginning, two blocks with a sun picture will open a path for you. Jump down and kill the skulls or keep going. Before going on, make sure you have a puppy with you. If you don't have one, head back to the Sand Zone Residence, talk to the mimiga at the far left, and take the dog next to the bed with you. Once you find a small entrance to your left, go in and open the door. Jenka will be inside, and you can give her the puppy (note that you can only carry one puppy at a time). this will trigger a plot point where Balrog comes in and asks for the key to the storage house with the flower seeds. when Jenka refuses, Balrog makes a threat and leaves.
Next, go find a block with a paw print on it to the right and up from Jenka's house. Go through to yield a secret passage, at the end of which is a life up and a treasure chest. open the chest, and find out that puppy #2 is sleeping inside. bring him back, and find the next one. head down from Jenka's house and walk across the sand (don't bother jumping the platforms yet). near the edge of this particular sandpit, the puppy will be there, standing next to a bone monster, like nothing's wrong (use the lvl 3 bubbler here, if you have it, this enemy jumps and the bubbler is good at taking out, even when he's in the air). when you get close to the puppy, however he runs away, but he's at a dead end, so he'll run right back, catch him with the down arrow when he does. return this puppy to Jenka now. next, go down to the same place as last time, but use the platforms this time.
midway though, there will be a place where you can jump up and find a hidden passage (marked by the paw print blocks again) jump up to find the puppy standing outside a door. predictably, the puppy goes into the door when you approach, so follow him. this area will be completely dark, but you can see a save point (use it), the puppy, yourself, the door, and some sand. avoid the sand, there's a croc in there, and you can't get to the puppy that way. if you have the map system, you can use it to see the general overlay of the maze, but if you don't, shoot around to find walls and gaps. when you get to the puppy, he starts barking his head off, so shut him up by picking him up. I repeat: do not step on the sand. it's easy to make this mistake on the way back. return him to Jenka, and go for the last one.
this one is a pain if you don't have the level 3 machine gun, due to the pits with instakill spikes, annoying jumping, and enemies which can be skipped if you could jetpack past them, but if you want the snake or the spur, persevere. there will be a giant throng of birds near the end. the best way to eliminate these ones if you don't have the machine gun is to use missiles to get them to get closer to you, then rapid fire with your bubbler (if it's level 3). Finally, you arrive at the storage area, but it is predictably locked. instead, go all the way to the right for a step up, then jump across the blocks to find a sleeping puppy. bring him along.
You will also see 2 breakable blocks nearby. Shoot them for a recharge station and a save point. go back to Jenka, and, surprise: Balrog beat the crap out of Jenka and has just taken the key. after a little dialogue, Jenka will tell you to go and stop them, and she also gives you a life jar. you only get to use it once, although if you do she can give you another, but I would suggest saving it for the final boss (note that if you want the best ending, the final boss is not the undead core, but Ballos, who is a very hard to beat and has 2-4 stages, depending on what you define as a stage).
 The Storehouse and the Power of the Red Flowers
Head back to the storehouse. the door will be wide open, but restore your health and save at the breakable blocks first. Entering the door will trigger a cutscene. Obviously, you are too late, and misery has already gotten the seeds for the doctor. You are not actually in the room at this point, so you can't stop them from doing what they do next. They want to test the power of the red flowers, so Misery summons Tokoro to use as a test subject. After a failed escape, Tokoro is grabbed by Balrog, who starts force-feeding her the flowers. Enter King. It is at this point that you see that king is both good-intentioned and awesome, because he comes in with a sword and takes Balrog out with one strike. however, you can see from Tokoro's new face sprite that King was also too late, and she has already eaten a flower. King makes one last attempt on the doctor's life, but misery zaps the crap out of him before he gets close enough. Enter you. The doctor and misery run away like wusses and leave you at the mercy of Enraged Tokoro (aka, Tokoro plus). Be warned, she is much more deadly than Igor, so stay on your guard (Frenzied Toroko). Note the lack of victory music after killing this once innocent Mimiga. Tokoro is gone, but king must have survived, right? Wrong. You can't leave until you talk to him, and when you do, he says some famous last words and then dies, leaving you his blade as a memento. Take the blade and leave. At this point, you hear the enthusiastic cheer of "Huzzah!" and Balrog jumps on you and smashes you into the sand. Misery comes back and teleports you to the labyrinth. and then, she does the same to Balrog.