When Olimar wakes up he inspects the damage to his spaceship and the situation is dire. Not only are the parts of his spaceship scattered over a wide area, but the Oxygen in the atmosphere is poisonous to Olimar and he must escape within 30 days, otherwise the batteries in his life support system will run out.
This part is a training area, there are no enemies to deal with so just explore and get used to controlling Olimar.
The Red onion
Eventually you will run across a dark red object which comes to life as you approach. It turns bright red and emits a seed which plants itself in the ground. You decide to call the object an onion since it resembles one. At this point you can control the camera and get a better look at your surroundings.
The seed quickly germinates and sprouts. Press 
next to it and you pull out a living creature. You call it a Pikmin after your favorite brand of carrots. The Pikmin starts to follow you around and you get a quick tutorial on how to control it. Press to throw the Pikmin; it will usually remain idle where it lands and turn a pale color. Press 
to call the Pikmin back. If you throw the Pikmin near something it can interact with it will attempt to do so. So if you throw it near a flower it will cut down the flower then return to you. The flower leaves a red disk called a food pellet behind and if you throw the Pikmin nearby it will start to carry it to the onion. When it gets there, the onion sucks in the pellet and produces two more seeds. These sprout and you get two additional Pikmin to follow you.
Explore the area for more flowers and stray pellets. There is a pellet at the top of a ramp to the left of the ship, also a flower and a larger pellet further to the left. The larger pellet is too big for just one Pikmin to handle. The game shows you how many Pikmin are needed to carry or move an object; the upper number (colored blue in the GC version) tells you the minimum number you need and the lower number (colored red in the GC version) shows how many are trying. You can have more Pikmin moving the object than are needed, and you get a small boost in speed from this, but there is a crowding factor as well and after a certain number more Pikmin won't be able to reach the object to help. For pellets the number needed to move it is written on the pellet itself, while the maximum number that can help move it is twice that amount. When the 5-pellet reaches the onion it produces 5 seeds instead of the 2 you get with a small pellet.
When multiple seeds sprout at once, you can save time by repeatedly tapping the button instead just once at each sprout. This causes you to autmatically seek out the nearest sprout instead of having to locate it yourself.
Using all flowers and pellets available at this point you should have 14 Pikmin.
Main Engine
There nothing more you can do now to increase the Pikmin population, so turn your attention to the cardboard box you may have noticed to the right of the onion. If you throw a Pikmin near it, it will attempt to move it and the blue number tells you need 10 Pikmin for this task. (This is also shown on the box itself.) When large numbers of Pikmin are needed for something, it's faster to direct them to it as group using 
; we'll call this swarming. Swarm the box and with 10 Pikmin working on it as a team they are able to move it.
With the box out of the way there is new territory for you to explore. The first thing you may notice is the Main Engine on the right. You still don't have enough Pikmin (20) to move it yet, so continue exploring. The box now forms a bridge leading to a large circular area. There are 3 small pellets and one large pellet here, so start your Pikmin to work collecting them. When the seeds are sprouted and plucked you should have 25 Pikmin to work with.
With no more pellets the only thing you can do now is get the engine. Gather all the Pikmin and swarm the part; they will start to carry it to the ship. When they get there the ship will absorb it and, thanks to rather spectacular auto-repair system, puts it its proper place. There is a cutscene showing you take off; the Pikmin go into the onion, which takes off as well and will now follow you wherever you land.
You can land here again but there isn't much point since you won't get any farther in exploring the area. So it's time to go to a new area which Olimar calls The Forest of Hope.
Return
Once you have all your Pikmin (Red, Yellow and Blue), you can grab the other part in the impact site. depending on what day you arrive, you will ether get goolix or mutana. once the mini boss is defeated the yellow pikmin will have to bombrock the wall to get to the part. Now you need the blues to attack the first clam in sight, as that one has the ship part. Meanwhile the reds and yellows will knock down a wall as a shortcut. the part should be carried by the blues and picked up by other pikmin, when the part is out of the water. soon the positron generator will be yours.