Poltergeist!
When you try to exit your room, you will be attacked by a lamp. Huh? Just keep attacking, and you'll have no trouble. In one of your sisters' rooms, you'll come across a doll. Again, fight it, simply beating it to submission. You should get a level up. CHECK the doll afterwards to hear a melody. Outside the house, you can get the basement key from the dog. Go into the basement and take all the presents. Now, head south to Podunk. The local enemies will initially be fairly strong, but after a few battles, they're not so tough. On the way you'll find a woman who is fretting about her daughter Pippi. Thus begins your first quest...
Rescue Pippi
Pippi is in the graveyard, taken by zombies. After you rescue her, you'll be given the Franklin Badge, which reflects PK Beam γ attacks. These are the only attacks it reflects (it doesn't even reflect other PK Beam attacks, only PK Beam γ), but it's good to have, because PK Beam γ kills instantly! You need only to have it in your inventory; you don't need to (indeed, can't) equip it.
Save the animals
Enemies near and in the zoo are tough, so be careful. You don't want to mess too much with tigers and elephants if you're running low on HP and PP.
You'll reach a capsule. CHECK it, and you'll fight Starman Jr. He will attempt to use PK Beam γ on you. The Franklin Badge will deflect this attack, saving you the trouble of fighting him. You can also use the rope to defeat him, as well.
Magicant
North of the zoo, past the bridge, you'll find a cave. Inside is what seems to be a strange-looking stalagmite. Inspect it and you'll hear a voice speaking into your head. Use telepathy. If you have your Great Grandfather's Diary in your inventory, you'll be able to answer the riddle. After you enter Magicant, you'll no longer need the diary, so give it to the Mysterious Goods Keeper.
Here is the only place you can buy armor. You'll find three shops, marked "7" on their signs (spelling out 777, apparently). Get the best armor from each: Magic Coin, Gold Ring, and Water Pendant (H2O Pendant). The Water Pendant is probably the best pendant, because some enemies burst into flames or explode when defeated, dealing out fire damage, while elemental PK attacks are uncommon until the end of the game (PK Beam isn't so uncommon, but it's not elemental).
North of the castle, you'll find a house with five Flying Men. You can take one as your companion -- you can't control him, but enemies will always attack the Flying Man instead of you. If he dies, you can take another, but remember that whenever one dies, he's gone forever. You can't heal them, either, so they probably won't last long. If you've been fighting all your battles so far, though, you don't really need them.
If you come across Groucho, and you don't have Flying Man with you, always guard. He will eventually say "Hello" and walk off, and one character in your party (just Ninten at this point, but you'll come back later with others) will receive a much larger increase in EXP than you'd gain by fighting. If you're with Flying Man, though, the EXP will go to him, which is useless.
Note: the Japanese version and the prototype English version of the game have a completely different layout for the final area.
Merrysville
Lloyd is hiding in a garbage can on the roof of Merrysville Elementary. You need to get a bottle rocket from the Little Sweet Factory, to the south, to entice him to come out. Buy him a boomerang, then use the Onyx Hook to go to Magicant so you can upgrade his armor. Go through Magicant again, leveling him up, and then you can proceed to the Duncan Factory.
Duncan Factory
A second Franklin Badge can be found here, which will be extremely handy near the end of the game.
Your goal here is to find a large rocket. After you CHECK it, Lloyd will aim it at the boulder blocking the way to Union Station, which is your next destination. Follow those railroad tracks!
Find Anna
On the train, go to Reindeer. Somebody at the station will give you Anna's hat. You don't actually need anything else in Reindeer, so go straight to Snowman. Be warned, the enemies in Snowman are very tough the first time you come across them. USE the hat while standing adjacent to Anna, and she'll join you. Now you'll probably need to use the Onyx Hook to get out with everyone alive, and you need to buy her armor anyway. When you get to Merrysville, buy her a boomerang.
A ghost house?
Note: the Japanese version and prototype English version have a different layout for this area.
Yucca Desert
Here you may have your first encounter with a nearly invincible enemy, R7037. Be sure to check the cactus with a face to learn the sixth melody.
Enter the dragon
If you are level 25, you can CHECK the dragon in Magicant to fight it. The dragon is very tough. If you want to do it the "right" way, then Anna needs to know SuprHealing (in case Ninten or Lloyd/Teddy falls). Use Defense Up β and PSI Shield β, and also OffenseUp on Ninten (and Teddy if you have him).
The easy way to defeat it is to buy the Super Bomb at Merrysville Elementary and have Lloyd use it. It will do a great deal of damage to Lloyd, but the dragon will be defeated immediately.
After you defeat the dragon, you will learn the sixth melody.
Youngtown
From the railroad tracks you used to walk to the desert, continue to follow them along until you reach the Youngtown Station. Continue heading south and eventually you'll reach Youngtown itself. Make sure you buy the Air Gun from the shop for Loid as it's his strongest weapon in the game, then check out the house on the western side (the one that has a baby in it). Use Telepathy on the baby and both Ninten and Ana will learn Teleportation, which allows you to visit any previously visited town you'd like, provided you have the space to use it (you'll see what I mean when you actually try to use it).
That's really all there is to do in Youngtown, so head to the east of town and cross the bridges you eventually come across. Soon, you'll find a house along the way. Its resident? None other than Pippi! Also, Loid's dad is outside the house in the garbage can, talk to him if you'd like. Thanks to Pippi being in this particular location, you may feel free to train to your heart's content as she offers free healing, making this quite a convenient little spot, provided none of your party dies, in which case you'll have to go to Magicant. Thankfully, now that you have Teleportation, getting back to Youngtown won't be any trouble at all!
Once you feel you're sufficiently leveled enough, continue along the bridges south-west from Pippi's new house and you'll wind up going east. Once you start to see trees, follow them north until you get to another bridge. In time, you'll find a bridge that goes west with a dirt path on the other side. Follow it and you'll find yourself in Ellay.
Teddy and the B.B. Gang
Mount Itoi
This is by far both the longest and hardest part of the game, even for experienced players. Mount Itoi has several "tiers" of enemies, progressing from tough to murderous. You don't want to fight all of them, or you will simply run out of supplies and PP, and you will be left unable to heal yourself and you will die. If everybody's around level 30, which they probably are, you're probably well enough off. (Don't worry too much if Teddy's a bit underleveled. He'll handle himself just fine.)
First you will bumble about in the caverns. In the first screen you can acquire the Katana, Teddy's best weapon. You can also find Hank's Bat elsewhere in the caverns. Likewise, you can find the Iron Skillet, but it is very marginally better than the boomerang (only 4 points stronger) and Anna is never going to be strong enough to physically attack anything by this point in the game. Make sure a Franklin Badge is in her inventory, so that she won't get hit with PK Beam γ (if others get hit, she can revive them). If you have a second badge, give it to whoever else you would rather be alive (probably Ninten or Teddy). If Anna falls in battle, you can use the Onyx Hook to go back to Magicant, heal up, and try again.
Once you get out of the caverns, the journey is only beginning. You will come across enemies such as Mega Borgs, which are not particularly strong, and Grizzly Bears, which are deadly. A grizzly can deliver 100 HP damage to a party member with full armor (you equipped everybody will full armor a long time ago, right?), and you don't want to burn PP needed to fight and heal the wounds. Run from them.
Eventually you'll find a house. Go into it, talk to the healer (he'll restore everybody to prime condition for free), save the game on the phone and only then proceed into the next room of the house. Ninten and Anna will be granted some privacy, which they'll use to start dancing. Then suddenly...
Remember R7037? Remember how you were screwed if you weren't in the tank? Well, you're not in a tank, and you're now facing R7037's cousin, R7038. Yep, you're screwed. There's nothing you can do, so do whatever you want. After your third party member falls, the tank will roll up and destroy the bot... it's piloted by Lloyd! (Note: you'll have your money halved because of losing the battle. That's OK, though, because money absolutely does not matter by this point.)
You then return to the house between Ellay and the caves. Sadly, Teddy is in critical condition. Lloyd takes his place -- and his inventory items. You can either start the long trek back up the mountain anew, or if you can't be bothered, you can intentionally die so that you go back to the house on the mountain (if you saved there).
North of the house, you'll come across a lake. Hang to its eastern side and follow it. You'll find a boat. Hop in (Lloyd will fix it), locate the whirlpool, and go into it. After a few screens, you'll meet the robot EVE. You'll then exit and be brought back to outside the lake. Do not attempt to walk back south or you will encounter R7038XX. This is an encounter you will be unable to avoid, but it's better to put it off because EVE will handle your battles quite well for you. You'll eventually come across a cave opening. Inside is the Sea Pendant, which defends against all PSI attacks. Give it to Anna, have her equip it, then just toss whatever pendant she was wearing (unless you want to give it to somebody else). Soon you'll meet R7038XX, and EVE will sacrifice herself to save you. CHECK the wreckage and you will find the seventh melody. (This is the second time Lloyd saved your skin: without Lloyd, you couldn't get EVE, and you would have been killed by R7038XX.)
You'll eventually come to a tombstone. (Giegue's lair lies just beyond, but it's blocked by a stone.) CHECK the tombstone and you will learn the final melody. You should have all eight melodies. In the original Famicom version of the game, you will warp to Magicant; in others, use the Onyx Hook to go there yourself. Go to Queen Mary and speak to her. After you sing the song, Magicant will fade away and you'll return to the tombstone on Mount Itoi. Walk north, and you'll see that the stone that was blocking the way will be cleared. You're ready to take on Giegue.
The final battle
Enter the cave. In the English prototype version, you will see a cave off to the right where you will see some prisoners of Giygas. (In the Japanese version, you'll come across them earlier on.) It's just for show.
Walk north -- make sure everybody has full HP, now -- and you'll come across the big guy himself: Giygas. (He's named Giegue in the English prototype, but see naming confusion above.) Giegue is easy if you know how to handle him, but extremely tough if you don't, so pay attention. He'll start with a monologue, with an incomprehensible attack with each sentence. Have Ninten cast PSI Shield β immediately (but don't bother with Defense Up β because it won't help). That's the key to this battle, because you'll suffer extremely heavy damage without it. After that, Ninten can successfully complete this battle alone if he has enough PP and keeps using Lifeup γ as appropriate, but it's better to keep other party members alive to get it done faster and more safely, especially if Ninten is low on PP. QuickUp on your characters might be helpful as well. Don't do anything else during Giegue's monologue other than healing, and reviving as necessary, guarding otherwise. Once Giegue stops talking, you can start singing. (In the English version, you can sing before then, but it will be ineffective.) Each time, the song will be more complete before you're interrupted. You will win the battle after singing the complete song.