This will just give you the answers. If you are just a bit stuck then I suggest you look at one of the hint threads. This is mostly a thinking mission so if you use this walkthrough it will be pretty boring. If you just want the reward though:
To start the mission you will need to:
-Explore at least 92 Sectors
-Play for at least 10 mins with the 2.1 patch installed
-Be in an Argon or Boron Sector
To do the mission you will need:
-A Mobile Drilling System or a couple of missiles which can destroy station-class asteroids
-A Mineral Scanner
-Cargo Life-Support System
-An Advanced Satellite
-A reasonably fast ship (you will be doing a lot of travelling) which can fight off some Xenon.
-Coordinates are given in km in the format x,y,z. To fly to coordinates go to the Autopilot Command Console (Shit+C) and select Navigation -> Move to Position. Use the numpad to select a position (the coords you have selected are in the bottom-right) and the press Enter. The autopilot will then fly towards that position. A lot of the hints have the coordinates at the end (after the name).
-To view the current clue you need to look at the mission breifing.
When you meet the requirements you will get a message saying the Yaki stole a Colossus. When you get this message, start looking in the pirate sectors around Brennan's Triumph for a station with the blue book icon (it's random). Talk to the Teladi and he'll sell you some information for 50,000 Cr. Accept his offer and follow the mission guidance to Weaver's Tempest and find a Susanowa Prototype called "Trouble Maker" with a lightbulb icon. Fly towards it and comm it a couple of times - you will be ignored. Your mission should then update telling you to destroy the Advanced Satellite and one of the Lasertowers. Do so and you'll get a message saying you can have the hint if you replace the satellite. Get as close as you can to the Navigational Beacon and then eject the Satellite from your cargo bay. You will then be given the first hint.
The hint wants you to go one sector across and five down from Kingdom's End. This is Ore Belt, and the next hint is inside an asteroid. The coordinates are given at the end of the hint (10,5,54). Head towards this asteroid and the mission will update. Blow it open with the Drilling System/Missile and collect the Data Storage Device. This gives you Hint 2.
The Foundation Guild HQ is in the sector description of The Hole. The answer is 4 jumps so it wants you to go to sector 4,3 which is Bala Gi's Joy. Head over to the coordinates given (25,1,50) and collect the Data Storage Device which is just floating in space.
Xenon Sector 347 (between Scale Plate Green and Eighteen Billion) is the one you want. There is a container at 90,0,0. When you get close to it, a load of Figher Drones (MK I & II) will spawn and when you destroy all of those the Data Storage Device will appear where the container was. If you're having trouble destroying them, you can just run away and wait for them to run out of energy.
The sector with four suns is Akeela's Beacon. There are seven Navigational Beacons in the sector. Six of them spawn Xenon and one of them jumps you directly to the Data Storage Device when you get near it. I think the beacons are positioned randomly so you'll just have to look around.
The anagram is 'Savage Spur'. The Data Storage Device is floating in space at 5,-30,67: behind three asteroids which are seperate from the rest.
The place where it all began is Seizewell (where you started in X-BTF). Following the directions from there reveals that the sector you want is President's End. Head over to the coordinates given to you (26,11,49) and collect the Data Storage Device.
You need to follow the sequence of numbers through the text - start with the 7th word, then go on another 5, and so on. If you do this it reads "Sector two jumps North of home of light" - so head over to Herron's Nebula and the Data Storage Device is at the coordinates given (0,50,0).
This hint refers to Rhonkar's Clouds. You'll just need to scan asteroids (with the mineral scanner) until you find the one with the Data Storage Device in it because the asteroid is positioned randomly, but luckily it's a small sector so this is much easier than Akeela's Beacon. When you find the asteroid, bust it open and collect the device just like you did in Ore Belt.
Finding the Ship:
Looking at the map in the mission screen, the green sector is Xenon Sector 101. There is ship debris at -10,10,-70. It should be fairly easy to spot as it's a pretty empty sector (apart from all the Xenon). Once you get within about 5km the mission will update telling you to scan it. If you get close enough (about 500m away?) to the ship debris, "scanning" will appear at the bottom of the screen. Wait for it to finish and the Xperimental shuttle will appear. You need to capture it (make the pilot bail) but this is not hard because the ship is invincible (can't go below 93% hull) so you can shoot at it as much as you want. Once the pilot has bailed, you'll need to collect him and take him to the Shipyard at Argon Prime (you can't claim the Shuttle so just leave it there). When you do you will be told to wait while they interrogate him. Just wait for a little while on SETA and you will get a message.
Because a few people are having trouble with this: when the Teladi attacks you, it can change your ship's IFF settings. If all Teladi have suddenly turned red, press shift+c, scroll down to the bottom and change Teladi back to friend.
You will get a message telling you to go and rescue the crew from a pirate TL. Follow the mission guidance to find it and then you have to keep its shields down and stay close (Less than 4km?) so the crew can transport over. The ship is indestructable so 'keeping the shields down' can just be blasting it with everything you have. Once all seven crew are in your ship, jump back to the Argon Prime shipyard and you will be given the Xperimental Shuttle (It is docked at the Shipyard).
Thanks to Carlo the Curious for the starting requirements and to Allan F, Vlad Zet, Ketraar and Graphil for coordinates and corrections.