Well the basic type Match-up is Grass, Fire, and Water. Here's how it works:
Grass Burns in Fire
Fire Depleats by Water
Water Gets Cut Up by Grass
But that's only for the simple ones. Comment if you want more added.
EDIT: I got more!
Fairy gets poisoned by Poison and Crushed by Steel
Steel gets shaken up by Ground, gets punched by Fighting and burned by Fire
Normal gets punched by Fighting
Psychic gets a nightmare by Dark and eaten by Bug
Dark gets punched by Fighting and sucked by Bug
Poison gets shaken by Ground and gets mental by Psychic
Bug is burned by Fire, eaten by Flying, and crushed by Rock
Ground is watered by water, frozen by Ice, and shaped by Grass
Flying is thrown off by Rock and frozen by Ice
Rock is shaken by Ground, broken by Fighting, and sliced by Steel
Electric is un-sparked by Ground
I have NO clue if I did then all, but that's reLlt all from Edit 1!