Instead of making a cuddly type kick Dragon ass, heres my take on the issue:

Dragon:
>Resists Fire, Water, Grass, Electric
>Weak to Steel, Ice, Dragon
>Immune to nothing
>Effective against Dragon

Fairy:
>Resists Dark, Bug, Fighting
>Weak to Poison, Steel, Fire
>Immune to nothing
>Effective against Dark, Fighting

New type: Light
Inb4 "Dark is called the Evil type in Japan". Light can be called the Good type.
>Resists Fire, Ice, Water
>Weak to Grass, Dark, Steel
>Immune to nothing
>Effective against Ice, Water, Dark

Steel:
>Resists Fairy, Light, Rock, Steel
>Weak to Fire, Fighting, Ground, Electric
>Immune to nothing
>Effective against Dragon, Fairy, Light

Ice:
>Resists Ice, Water, Flying
>Weak to Fire, Light, Fighting
>Immune to nothing
>Effective against Dragon, Flying, Grass, Rock

Dark:
>Resists Psychic, Ghost, Normal
>Weak to Fairy, Fighting, Light
>Immune to nothing
>Effective against Light, Psychic, Normal

What do you think?