Step by step summary:
1.Obtain your Tome from a Berzerker Tome vendor in your home city or in Plane of Knowledge.
2. Give your Tome to your Guild Master in your home city or in Plane of Knowledge. You will get a "... use your skills wisely..." message.
3. Press the Atl and C keys at the same time to bring up the Combat Abitlites window. Click on the "S" to get a list of abitlites that you have turned in the tomes for. Highlight the ability and you will get the option to make a hotkey for it. Make a hotkey either on the Combat window or on your HotKey Window or both.
4. Read the message from your home city Guild Master and you will be able to tell what nearby NPC is selling the Basic Axe Components. Example : the guy just outside the bank in Kaladim
5. Purchase the Basic Axe Components.
6. Now, click on your new hotbutton. Example: Blunt Axe. You will summon a stack of 20 Blunt Axes. These can be thrown if in your Range or Ammo Slots. Important note: The Blunt Axe is also the 'reagent' for the combat abilitles Leg Strike (snare) and Head Strike (stun). Clicking on the buttons you make for Blunt, Corroded, Bearded ...etc makes more axes, it does not employ the axe as a weapon.
7. You will eventually use the other axes for throwing as you advance in levels, but you will always want to have several stacks of Blunt Axes. When used as reagents, they do not have to be in a range or ammo slot, but in a backpack.
Hopefully, this has cleared up some of the confusion.