You begin each stage of Rainbow Islands at the bottom, and your goal is to climb all the way to the top where it says "GOAL IN!". To do this, you must either jump from platform to platform, or create rainbows and climb on top of them to get higher. However, Rainbow Islands is much more complex than it immediately appears.
Basic island description
When you begin a new game, there are seven islands, and each island is broken up into four separate vertical stages, for a total of 28 stages. However, there are three additional islands which are initially hidden. You must perform a particular task on each of the seven original islands in order to reveal the three hidden islands. These three islands bring the stage total up to 40.
On each island, you must complete the four stages by climbing from the bottom of the stage to the top. Each stage of an island is typically taller than the previous stage. Along the way, you must dodge and/or attack the various enemies that occupy the stage. When you reach the very top, a treasure chest will fall from the sky, open up, and deposit a number of items around for you to collect. These items will always be worth 500 points regardless of how much they might be worth if you collected them during the stage.
At the end of the fourth and final stage of each island, you must do battle with a boss. Each boss has a health meter. That meter can be depleted by attacking and crushing the boss with rainbows. Once the boss is defeated, another treasure chest will appear. In addition to the regular 500 point items, one large 100,000 point item will appear from the chest.
If you defeat the boss of the seventh island, but fail to make the three hidden islands appear, you will receive the normal "bad" ending to the game.
In order to make the final three islands appear, you must collect the seven large diamonds from each of the original islands. In order to collect a large diamond from each island, you must manage to collect all seven small diamonds before reaching and defeating the boss. If you possess all seven small diamonds when the boss is defeated, the treasure chest that appears will contain a large diamond instead of another large food item. You will be able to see which islands you have collected all of the diamonds for when you see the island map.
Continuing the trend
Once you make it to the final three islands, your effort to collect the seven small diamonds does not end. Instead of earning large diamonds after defeating the boss of an island, you must collect the three mirrors that are needed to earn the true "good" ending. If you do not have all seven small diamonds when you reach the boss, you will not be able to collect one of the mirrors, and thus not be able to see the true ending.
The silver doors
It's one thing to make a large diamond appear when you defeat the boss of an island. It's entirely different to make a silver door appear. Silver doors appear at the top of the screen when the boss appears. In order to make a silver door appear, not only must you collect all seven small diamonds, but you must collect them in order. That is, you must collect them from red to violet, or from violet to red. If you collect any diamond out of order, you will not be able to access the silver door at the end of the island.
If you manage to collect all seven small diamonds in order, and make the silver door appear, you can jump through the door at any time during the fight. Passing through the silver door, you will find the large diamond or mirror for that island, along with a giant power-up. The effects from the giant power-ups found behind the silver doors are permanent, and will last until the end of the game. The only exception are the gold doors found behind the silver door of Island 5. These doors allow you to warp ahead to Island 7 or Island 8. If you wish to see the true ending, you must avoid using either one of these doors.
Earning tremendous points
The presence of the silver doors makes it possible to earn an incredible amount of points. To do this, you must wait to jump through the door until after you have defeated the boss. The treasure chest will drop down and release the large diamond or mirror. Collect the 100,000 point item and then race up to the silver door before the game advances you onto the next island. Once through the door, you'll be able to collect the same item again for double the points.