From Bulbapedia, the community-driven Pokémon encyclopedia.
After the game shows a small scene of a Pikachu, Meowth, and Munchlax all in hot air balloons, you'll be taken to the title screen. Select the "One-Player" balloon.
The game will then start a tutorial on how to play the game.
At the top of the DS, there will be a map showing you who is ahead.
The first thing taught is the sliding action. To do so, simply slide your stylus on the touch screen. If you want to change how you slide, you can select the way you prefer at the Options balloon.
The "checkpoints" are things on the ground that look like Poké Balls. You can go wildly around and collect them--however, the only way to seriously complete the race is to collect them in the right order. Step on them in order to activate them. The direction of the next checkpoint is shown by an arrow on top of Pikachu. The number of checkpoints left will be determined after you activate them. You'll be racing against Munchlax in all tutorials.
The color of the arrow on top of Pikachu shows you how far the next one is. Pink=closer, yellow=close, green=far.
After collecting the last checkpoint, you'll finish the first part of the tutorial and go to the next section.
The next thing the game teaches you is the different landscapes found on races.
When crossing forests or sandy areas, your speed will decrease. As for the ocean, you must stay away from there or you'll drown, which will cause you to slow and the other Pokémon to go ahead of you. Avoid the ocean as much as possible.
Swamps, on the other hand, you are allowed to cross, but try to make it across as quick as you can. If you stay there for too long, you'll drown.
Step on a pad to run faster in certain areas, such as stepping on the Forest Pad will make you faster in a forest area, and stepping on a Sand Pad will make you faster on a sandy area. As for going across swamps, likely, there will be a Swamp Pad that prevents you from drowning in a swamp, no matter how long you stay.
To go across the ocean, you'll have to ride on a Lapras. The ocean here isn't too long, however in future races the oceans will be longer.
Hot-Air Balloon and Radar
When you run forward, you'll realize you are required to cross the ocean, however there is no Lapras to take you for a ride. However, go and grab some balloons, which will trigger some arrows to appear on the side of the touch screen. When you use the balloons, you can fly high in order to cross paths you wouldn't normally be able to without some special power.
To use the balloons, slide up or down on the right or left side of the touch screen.
By looking at the radar at the top of the upper screen, you can check your next checkpoint from the sky. To land with your air balloon, you use your shadow to decide where you're going to land, and slide on DIVE with your stylus.
This part of the tutorial tells you how Pikachu can't land in certain areas, like a forest. If Pikachu does land there, it'll go back into the air.
You can pop a balloon when Pikachu is going to land in order to go to the ground faster. However, there is a downside--if you pop the balloons too fast and land on solid ground, you will crash, land...and faint. Roads/rocks are solid, and icy areas are a bit solid.
Since you finished the tutorial, you can now challenge the Grand Prix.