but I'm stuck figuring out how to take control of a colonie that leaves you. Or how to keep a planet from leaving your side?
Sounds like culture. You have to build your own culture centers (Which require research.) or station your capital ships (They can push back enemy culture, but can't produce any of your own.) at your worlds. There's also some upgrades that can be researched to improve culture spread and give you a race-specific bonus.
How do i influance them and also influance the computer AI to join my cause?
Check the diplomacy screen to see what missions the AI has given you. If you complete the missions, the happiness level will go up and they'll be more likely to offer/accept agreements.
