Build a colony ship, take it to one of your planets that has a population. Then put it in orbit and eject it from orbit. You will be asked how much of the population you want to load on the ship.

Now fly it to a planet thats habitable and put it in orbit and you will be asked to name it. Da dah new colony. You can build these ships from the start of the game.

With your mouse left-click on the colony ship to select it. Right-click on the planet (or any destination) you want to send your ship to. When a colony ship arrives at a suitable planet you will be prompted whether you wish to colonize it.