Transport Boats do not need to be researched, you can build them right away if you have a harbour. They are the main way to transport troops from one island to another. They do not fight and should be protected by other ships when attacking a city.
Light Ships are strong attacking ships. They are the main offensive ship in the game. You should always send a Light Ship escort with your transport ships when attacking.
Fire Ships are ships that can only be used for defense: they are the last defense and will only enter into battle after the regular naval battle. Fire Ships can not destroy Colony Ships or Transport Ships. Each Fireship can destroy on other ship, however it is also destroyed when it does so.