Ukkos who guards the dockside warehouse in Harbortown sells Axe 2 skill training for 200 gold and 10 learning points. He sells Axe 3 for 300 gold and 10 learning points but I dont know how far he trains in axe as I hadnt enough training points to proceed further.
The default level allows you to wield an axe, attack, parry and combine up to 3 attack blows.
Skill level 2 adds the ability to charge your axe attack for increased damage or to break through an opponents parry.
Skill level 3 allows you to perform "lateral blows" which allows you to "knock your enemy to the side with your axe, push them and you can hit them again" according to Ukkos the Ogre.
Skill level 4 allows you to carry out counter parries to interrupt your opponents attack. (allowing a riposte)