I am on an island in CIV5 but it looks like my ships are stuck
In order to travel the "dark blue" ocean tiles, and escape the limits of the "light blue" coast, you'll need Astronomy. Astronomy gives you the Caravel, a fast scouting naval unit that can cross ocean tiles, as well as allowing embarked units to travel on these tiles as well.
It's often the case that when you start on a remote island, it may be difficult or even impossible to find another landmass without Astronomy.