Constructors do not need to phase jump. When you colonize a planet, the constructors are automatically spawned for you. To build stuff, always click on the planet to issue build orders, never give orders through the individual constructor (you can, but it takes to long to find them 
)
Only ice and volcanic planets need research to colonize. You will need two civic labs and can be found in your noncombat tree. The first planet you should colonize is a moon or asteroid, which most of the time are next to your homeworld.
Also, are you playing trinity or Rebellion? If trinity, I would try a game wtihout any AI players first. If Rebellion, I'll be updating my map that strands the AI at their home world soon.
