Traverse Town

When you regain control after the short cut-scene, proceed to the accessory shop and you'll meet Cid there. Jump on the cabinet for a chest with a Mythril Shard. Leave the shop and head east up the stairs to the Second District.

The town will also be infested with Heartless, just fight your way through. Feel free to explore a bit to familiarize yourself with the area. Once you're done, go back to the accessory shop at the First District and save your game. Leave the shop. You'll meet Leon here. There'll be a short cut-scene between him and Sora about the keyblade. You'll have to fight him after that. What a bully.

Boss Battle - Leon (Squall)

Avoid his trigger attacks at all costs, it's real deadly. He'll also shoot fire spells at you, but that shouldn't be a problem. I found him most vulnerable after he does a combo. Now when you get close enough, try pushing him back into a corner with your own combo attacks. Once you've done that, just stay close and smash away to prevent him from striking back. You'll win after a while. It won't be game over even if you lose to him. You just don't get any experience points.

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Watch a long and interesting cut-scene. When you regain control at the hotel room, jump onto the table and examine the treasure chest (it'll be locked). Now talk to Yuffie and she'll tell you that the keyblade can open locked doors and treasure chests, so examine the chest again to get the Elixir inside. Save your game and talk to Leon. Tell him you're ready.

There will be another cut-scene where the Heartless attack the town. (as if they hadn't already) You'll regain control at the alleyway. After giving a not so useful advice, Leon will leave you to fend for yourself. Here, go south then north at the corner for a chest with a Potion. Head back to the Second District afterwards. The Third District is your next destination, but feel free to go back to the First District if you need to stock up on items. I suggest that you go save at the accessory shop too. Be sure to talk to Aerith there for a Mega-Potion.

At the 3rd District, you'll see a short cut-scene. Donald and Goofy will join now. Just defeat all the Heartless here and a massive piece of junk will appear out of nowhere.

Boss Battle - Guard Armor

This one has a lot of parts, therefore make sure that you focus on only one target at a time. The feet are the easiest ones to hit so get rid of them first. Once they're gone go for the hands then the body. Watch out for its devastating spinning attack though. Killing each part yields HP balls which helps quite a bit. Don't worry too much about healing -- let Donald or Goofy do that for you.

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After the battle, there'll be some long cut-scenes. Then you'll be back at the 1st District where Aerith, Leon and Yuffie will give you 100 munny plus an Elixir (if you managed to beat Leon). Donald and Goofy will teach you the Fire magic and Dodge Roll ability respectively. Proceed to the accessory shop to get the Blue Trinity. Go back to the Hotel. You can now access the Red Room, get in there for a chest with a Pretty Stone. Now proceed to the Green Room and hit the clock located above the door to these times: 12:00, 6:00, 7:00. This will reveal a chest with a Mythril.

Once done, go through the door that leads to the Alleyway. Here just go east to the end for another Pretty Stone, then west for a Potion.

Lastly, go to the 3rd District and use Fire magic on the door with a flame marking on it to access the Magician's Study, where you'll meet Merlin and the Fairy Godmother. Activate Blue Trinity marks if you wish. When you're ready, leave the town through the World Exit. You'll have some tutorials about the Gummi Ship. You can now go to 3 new areas, which you can tackle in any order.