This walkthrough is for v1.0X. It may not be viable in v1.1 and/or on high difficulties.
First turn:
Design a new constructor with 1 constructor modules, 1 impulse drive, 2 general life supports. Design a new colony ship with 1 colony module, 2 impulse drives, 2 general life supports. Design a cargo hull fighter with atleast 2 general life supports, atleast 2 impulse drives, and atleast 1 laser.
Move your colony ship into orbit and relaunch with 500 population. Move all your ships left full movement. Upgrade all your constructors to the new design, and upgrade your colony ship to the new design. Buy a lab on your capitol and starports on the two planets that don't have it. Buy three of your new colony ship design.
Set spending to 100% and Research to 100%. Begin researching planetary invasion.
Turn 2:
Launch your 3 new colony ships with full population. Buy 3 of the fighter design. Buy a lab on each planet. Move all ships left full movement.
Turn 3:
Launch you 3 fighters. Buy 3 more colony ships (to be safe). Buy 3 more labs. Move all ships left full movement.
Turn 4+: Keep moving your ships left. Try to colonize the planet in the middle of the map if possible. If you reach a limit on your range, use a constructor to build a starbase. When your fighters arrive at the drengin planet, take out any ships they have. When you research Planetary invasion, design a new troop ship that has atleast 2 general life supports, 1 troop module, and engines. When your colony ships reach the drengin planet, upgrade them to the new troop ship design (you may have to wait for the tech to finish).
Invade and win!
This took me less than 20 turns (i didn't count) on the default difficulty. Most all of that was moving across the map.