When you build a new ship, it starts orbiting the planet where you built it. You have to "launch" it from the planet in order to be able to move it and other stuff. By clicking a planet you should be able to see the ships orbiting it, and launch them accordingly.

When you click on one of your planets you should see the planet´s details in the central area below the main galaxy view. There you can see the planet´s population, approval, influence points, etc. To the right of the planet´s details you should see a small window showing what ship the local starport is building (if any). There you can also see what ships are orbiting the planet, and there´s also a button for launching any of them (you have to select which one to launch first).

Hope this clears it up!

Edit: Here you can see it (though without any ships orbiting the planet)

If there was any ship orbiting the planet, you´d see small icons of them below the "Ships in orbit:" headline.