When you come across 3 pink birds in a diagonal line you'll notice a ledge below them. Approach the right edge of the ledge and have Kid Niki just have his left foot on it. Press down and B repeatedly to make Kid Niki jump into the air, flash, and warp to a bonus area that'll take you close to the end of the level.
The beginning of the second level has you walking through a forest. After a while, there will be a gap in the trees, the Round meter should read 2-6 at this point. Climb the first tree after this gap, and when you reach the top, you'll be taken off to a bonus area. Swing your sword around in the room you arrive in to get coins for bonus points.
Throughout level 3 you'll come across small cloud platforms that are bobbing up and down. Make your way to the third one of these. If your Round meter reads 3-7, you're in the right place. When the platform reaches its highest point, jump. Continue to do this, and on the fourth time, you should be taken to a bonus area.