Beginning
Abilities Albel knows from the beginning
Air Slash
Shockwave Swirl

Learn
Abilities Albel will learn by just leveling-up.
Level 21 - Berserk
Level 25 - Stun
Level 26 - No Guard
Level 26 - Palm of Destruction
Level 29 - Aerial
Level 32 - Aura Wall
Level 32 - Charge
Level 33 - Fear
Level 35 - Double Slash
Level 40 - Vampire Slash
Level 45 - Hand of Doom
Level 50 - Dragon Roar

Other
Abilities Albel can learn in a different way
Air Slash of Fury:
Learned: From book (Cave of Trials, Floor 6)

Double Slash of Fury
From book (Sphere 211, Floor 135)

Symbology?
Albel doesn't learn Symbology at all, unlike characters like Sophia, Nel and Fayt. However, you can teach him 'common attack symbols' (Fire bolt, Ice Needles, Earth Glave) and 'common support symbols' (Healing, Silence, Antidote) by the tomes you'll find pretty early in the game. It comes in handy when you need a healer.

Albel's Battle Skills in Japanese/English: Additionally, I decided to include this. It comes in handy when you're looking in a walkthrough/FAQ and it has japanese names.

Kuuhazan - Air Slash
Shouretsuha - Shockwave Swirl
Kikoushou - Palm of Destruction
Mashouseki - Aura Wall
Souhazan - Double Slash
Makousen - Vampire Flash
Goumashou - Hand of Doom
Kouryuuh - Dragon Roar
Mugen Kuuhazan - Air Slash of Fury
Mugen Souhazan - Double Slash of Fury