World 2 isn't much harder than World 1. At the start, there's a crate to the right, and a few jumps over some water. The crates may conceal a coin, but none past the second body of water do so.
Of note is the first hint door, found in the top-right corner of this world, which says "Secret switch in cave might create doors". You can skip the secret area by continuing to the exit.
By the time you reached that, you most likely saw a large cave. Enter the large cave first, kick the treasure chest until you obtain the little player. With the little player, have to the smaller cave, and send him below. At the end of the small cave is a switch, which reveals a door at the start of the map.
World 18[edit]
World 18 is a secret level accessible from World 2. This world uses the clouds as platforms.
Of note, there is a secret switch at the bottom-left corner, and additional hidden switches found above the three buildings (which requires a crate to reach).
When you reach the treasure chest, you can jump on the clouds leading to the left to collect additional coins.
To continue right, you will have to use the clouds as platforms. This leads to a section where there are cloud-platforms moving along the track before reaching the exit.
At the exit, there is a hidden switch to the right of the key. Flipping it reveals an extra life, which you can take before returning to World 3.