you will discover these things as you continue to play the game!
but for now a quick explanation:
Convertible - if a piece of gear is convertible it means that once it is 100% spiritbonded it can be converted into materia.
Projectible - means it can be glamoured onto another piece of equipment if the appropriate conditions are met
Desynthesizable - once you unlock this ability you can break down this equipment into baser components
Crest-worthy - if you are in an FC you can choose to have your FC crest appear on your equipment

You are a THM and you started in Ul'Dah - keep going on in the Main story and eventually you will gain the ability to travel to other city-states. Once you reach Limsa Lominsa then you can get to the Arcanist guild and sign up to become an Arcanist. The ferry from Vesper Bay will become accessible eventually as well. Just keep playing on! Good luck!