Extra Life
At the Bubble Level, go towards the bubble and get in it. Once in the bubble, go to where you started and go up all the way. You will now have an extra life.

Easy Mode
At the Title screen, press Start to go into the Pipe Room. Then press Select to enable Easy Mode. You can now start any saved file under Easy Mode.

The 6th Coin
Choose to go to the Hippo Stage. There, you'll see a model of a hippo blowing bubbles. Jump into a bubble and float all the way to the end of the level. Instead of exiting through the regular door, exit through the door above it. You would not have been able to reach this door if you weren't in the bubble. After exiting through that door, you will have to play a hard space stage. If/When you complete the level you'll get the 6th coin