The Forest Naval
Clearly the easiest one to get. All you have to do is simply head South from your shop with about 30 Pikmin until you find a bundle of sticks. Get your Pikmin to build it into a large stick and they will climb across and bring it back to your ship. It might be wise to head around first and start taking down walls. All you need is 15 to carry the piece though.
If you've been watching the map, you may have noticed one of the pieces moving around. That's because one of the creatures in The Forest Navel has it. It likes to steal things, and as a matter of fact, that's how you kill it. When you see this snail like creature, find a flower and kill it until the pellet is on the ground near the creature. If he spots it he will run over and try to take it. Just throw two Pikmin at it, or however many it takes, and instead of heading back to the creatures home the Pikmin will force the creature to the Onion, once he gets sucked up he will be killed and the Space Float will be left.
#1 Ionium Jet (blue Pikmin required)
Once you've found the blue Pikmin near the water, use the pellets around them to make 15 of them or more. Once you have, check your map. In the pool just North of where you found the Blue Onion (Use the red arrow on the map to see which way is North) you'll see a ship piece. Head there with the 15 blue Pikmin which can survive going underwater, and take it back to your ship.
Anti-Dioxin Filter (blue Pikmin required)
Head South West down to the lake with a team of red Pikmin and take out all those fire breathing baddies. Watch out for the frogs. The best way to take the frogs down is to wait for them to drop and then surround them. Call the attacking Pikmin quickly back, when he jumps back up he knocks them all to the ground and then stomps on them. Then once that area is cleared bring your blue Pikmin down into the lake, clear out the nearby frogs and take the Anti-Dioxin Filter back to the ship.
Gravity Jumper (blue Pikmin required)
Head down to the same lake as mentioned above with a team of blue Pikmin. At the very back of the lake you'll find some land and some sticks to build a bridge. Build the bridge, and when you get up don't pick the piece up just yet, build the other bridge, then have your Pikmin take the Gravity Jumper back to the ship.
Libra (yellow and blue Pikmin required)
Assemble a team of Pikmin, 25 of each color. Head down to the lake, then head up North from there, to the only path marked on the map. It's the only way to get there so it shouldn't be to hard. Make sure all those fire breathers are dead, and if they aren't use your red Pikmin to kill them. Then take care of the black wall and head down the path. Make sure your Pikmin hug the wall or they will get burnt. Once you get to the bridge have your Pikmin get to work on it. When they finish building it, take your yellow Pikmin and throw them up the ledge, and take your blue Pikmin and pop the geyser in the water. Then jump on the geyser. Then throw the yellows all the way up to the Libra and let them haul it down. Once they get to the bottom let the reds take it back. The fire along the way will kill the other two kinds of Pikmin.
Analog Computer (blue Pikmin required)
Continue on up the path from the last ship piece. You'll run into the Analog Computer in a pool of water. Use your blue Pikmin to take the Analog Computer out of the pool, then switch to your reds and let them finish taking it back, as they are the only ones immune to fire. If you don't have enough reds, and you won't if you've been following all the way through, head up to the upper ledge, where you'll find a flower that's red. Throw however many yellows in there you need to, they will all turn red. Then haul the piece back to the ship.
Guard Satellite (yellow Pikmin required)
This is the piece all by itself in a room to the East of you. Make a team of 50 red and 10 yellow Pikmin. Take them all and head East until you find a big black door, and make sure to kill all the bugs there. Then get the reds started on knocking down the door. Meanwhile, take the yellows and head to where you found the Gravity Jumper, and you should find a bucket with some bombs in it. Get your Pikmin to take the bombs. Then head back to the big black door but head to the opposite side that you were last on. You should find a stone door that needs to be bombed, so you know what to do. After your finished bombing that door get your yellows to help with the big black door. Once its taken down head inside. This is the next boss. Just dodge around his feet until he lowers his center body and then throw all your Pikmin onto that center round part. May take a couple of tries before you can do it without losing a ton of Pikmin. Then haul the piece back to the ship.
Head South East from the ship with a team of a bout 50 red Pikmin. Your about to fight the mushroom boss. He's actually pretty easy if you know the C-stick throw technique. Just head down there, and take out everything on the way. Once you see him, get close to him and start throwing Pikmin at/on him. If he sparkles run away, or your Pikmin will turn purple and start attacking you! Just wait till he flips over, exposing his legs, then use the QuickThrow technique to launch all your Pikmin on there, and you should be able to kill him in one go. Then just take the stabilizer back to the ship.