You will be able to collect a speed potion early on in this world. In addition to the first one you collect in World 1, this should be your second, which means you only need one more potion to reach maximum speed. That third potion can be obtained if you take the right branch off the main track and head north. Once you collect the second speed potion, double back since the only thing you will find further north is a warp zone to World 4.
Once you're back on the main track, continue moving north to collect both the Rock Case and the Artificial Heart for this world. Once you have both, double back until you can travel to the right. Then continue north up this column of screens to find a Power Rock, and a Konami Logo. Collect the logo to make Konamiman appear. If you catch him, he will restore all of your health to maximum. Finally, head up to the boss room to challenge the giant spider for the key.
The boss of World 2 is a giant spider. It attacks in a very similar manner to the boss of World 1, by sending out many smaller spiders minions. In addition, it also shoots webs at you. If you are hit by these webs, you will become temporarily immobilized. Even if this happens, do not stop throwing rocks at the spider unless you are unable to hit it with them.
Once the boss is defeated, you should return to the very bottom of the World where you arrived from World 1. The right door contains the warp zone that leads to World 3. Two rooms below the Boss Room is an in-world warp zone that can make travelling back to the beginning much faster than walking there.