There is indeed visible skill progression for most crafts too. Current craft skill levels can be viewed under Main Menu 1 -> Status -> Craft Levels, alongside your Combat Skills and Magic Skills.
You are a long ways off from it right now, but keep in mind that unlike combat/magic skills, crafts have a limitation. You can freely skill up any craft up to 60. After that, you are only permitted 40 levels to divide amongst any of your 60+ crafts. In short, if you raise one craft to 100, all your others will be stuck at 60. If you then try to raise one of those, your 100 craft will start to delevel. HOWEVER, fishing is an exception to this rule. It is possible to have 100 fishing and 100 in another craft.
Also, crafts cap at every 10 levels. Once you reach within 2 levels of this cap (8, 18, 28, etc.) you must travel to the guild of your respective craft and submit a "test" item (generally, it's an item that can be produced using the craft) in order to break the cap and allow you to pass that 10th level.
There is one "craft" that does have a form of skill progression (chocobo digging), but it is invisible to the player and can only be determined by carefully examining certain aspects of your chocobo's digging. It is more of a HELM (resource gathering) type of activity than a real craft, though.