This might be a bit confusing, especially if you're like me and NEVER tinkered with the Gummi ship because of how complex it was. Sadly those days are over :(
When you're at the "Select World" screen you can press the Square button to visit your Gummi Garage or press Triangle to see your missions.
By default the game will show you the current mission for the world you're currently on, showing your overall score, what mission number you're on (between 1 and 3) and the level of difficulty (1 to 5 stars).
Every time you complete the first two missions in a world you'll get new Gummi pieces but the 3rd mission rewards you with a new blueprint.
The idea is simple enough: complete the task, following the instructions as per the mission summary, and END ON THE WORLD YOU'RE TRYING TO COMPLETE THE MISSION FOR.
In my case I did the Deep Jungle challenges so I'll use that for this guide as it's the easiest imo but to reiterate, no matter what I have to do for these missions it only registers *IF* I select Deep Jungle as my destination and I CANNOT use the Warp Drive!
Mission 1: Obtain a score of 200 or higher
This is super easy as all you need to do is add a ton of weapons to your Gummi (new or old it doesn't matter). Everything you do gives you points so you can kill 200 enemy ships, collect 200 Special orbs from meteors, collect 200 shields, or any combination of the drops. As long as your final score is 200 or more when you reach the Jungle you'll be fine.
Misison 2: Start in Agrabah and collect 10 Special blocks.
This is a little tougher because you can't just go from Agrabah to Deep Jungle directly; you need to use the warp hole.
Select Agrabah as your destination from the map (you can warp to it if you need to because this is your STARTING point and we're going to the Jungle!) and once you arrive just select "Select World" from the list of locations.
From there select the orange Warp Hole and drive through the game as normal, being extra careful to shoot down the meteors that fly around as those are the only things that drop Special blocks that you need to drive into to collect (they look like stars).
After about halfway you'll see flashing bars on the screen telling you there are two directions to go. PICK THE RIGHT SIDE OF THE SCREEN! AVOID THE LEFT!
make it to the Deep Jungle and you're down to 1.
Mission 3: Start in Wonderland and shoot down 55 ships using only Thundaga weapons.
THIS. IS. RIDICULOUS.
It isn't hard, per se, it's just a pain collecting Thundaga weapons! Thundaga weapons are missiles so there's 2 or 3 seconds between shots meaning that you can't spray and pray.
The best advice I can give is to wait for enemies to cluster up like when thy're going to fly into you before you open fire. Also the ship I used for this was a custom build that was just a cockpit, the Holy-G engine, and 4 Thundaga -G weapons, two directly under the cockpit and one on either side.
55 enemies may seem like a lot but when you're getting 5 or 6 in a cluster and bombing them to Kingdom Hearts it's pretty easy to rack them up.