Starbound saves your progress in two different folders; players and universe. Both of these are located in your main Starbound folder (in linux32 or in linux64 if you use Linux version).

• Starbound Custom Ships
• Starbound Ship Upgrades

I've found that if I create a new character while in the Multiplayer screen and then go to a server right from there, it works fine. If I go into Singleplayer for any reason with the new character, then I am unable to go into servers after, getting the same 'Client ship world has errored' message.

Starbound Custom Ships

Starbound Ship Upgrades

The players folder stores your character, your inventory and your ship. The universe folder stores all the planets you've explored and any changes made to them. If you want to transfer everything to another computer, just copy the content of both of these folders and paste it in the same location on the other computer.