Go through the tunnel ahead of you, jumping over the lava pits and collecting crystals. At the end you will find a suspended chest with a golden key on it. Grab it and stomp the crate to get some crystals. Then pass the two lava pools on the left using the collapsing platforms and killing the enemies you meet. Then go through the new tunnel until you reach a door.
This part is very linear. You will simply have to follow the tunnel, passing the lava pools, defeating some simple enemies and collecting crystals along the way. To overcome the first pool you will have to use an elastic platform, while to pass beyond the third (and last), you will have to jump with timing on a platform that appears and disappears. Nothing complicated. Enter the door at the end.
In the next cave, jump to the platforms above the lava pool. The last one, when you climb on it, will move upwards, allowing you to climb to the upper level. Here you will find the puzzle piece, but before you get it, open the door on the left with the golden key and enter.
You will enter a very high cave, with a long row of platforms arranged in steps that rotate around the central pillar. Climb these platforms and at the top you can get two extra lives. Then let yourself fall and return to the previous area. Take the puzzle piece to complete the level. That's it, for the first island.