I didnt play for a while, but i think there is a "Wooden axe" or somethin like that under carpentry? At least for one handed weapons etc, there are trainingsweapons there. Normally they need a billet or two and a Rope.
Yea that is correct, you will need to craft a Practice Great Axe, which required a carpenters toolkit and some level of skill in carpentry if I recall correctly. The most difficult ingredient to acquire is a rope.
All weapon skills below 30 can only be raised with the associated training weapon.