How to transfer universe to another PC?
Solution 1:
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).
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.