Dual types can also be read from the main chart, it is a bit more complicated, but surprisingly effective once you get the hang of it.
Defense
Find both your pokemons types in the chart.
All weaknesses to both types (coming from the left)
apply to the pokemon if it is not "cancelled" by
a resistance on the other type.
Same is true for resistances, all resistances applies to the pokemon if it has one on either type, if its not "cancelled" by a weakness from the other type.
Azumarill
Water & Fairy
Weak to (+60% dmg)
Grass, Electric & Poison
Strong against (+60% dmg)
Fire, Ground, Rock, Fighting, Dragon & Dark
Double Resistance
If you have a resistance type in common on both your types, you will have a double resistance against that type and take 61% less dmg(same as for immunity) Look for the same color on the resistance bars with the shields.
Resists
Fire, Water, Ice, Dark, Fighting & Bug (38% less dmg)
Dragon (immune) (61% less dmg)
Strong against (+60% dmg)
Grass, Fighting, Bug & Dragon
Altaria
Flying & Dragon
Weak to (+60% dmg)
Rock, Dragon, Fairy
Double Weak to Ice (+156 dmg)
Resists
Grass, Fighting, Bug, Fire, Water (38% less dmg)
Ground(immune) (61% less dmg)
Double Resist Grass (61% less dmg)
Attack
Attacking side works a bit more simplistic. If your pokemon has an attack of a certain type it will deal super effective damage to that type (to the right). Your pokemon might have two types but your attacking move has only one type, so only read the type that applies to your move.
As explained above it will not be super effective if the pokemon resists it on either one of their types. It is always the defending pokemons types that can complicate things with double weakness or double resistance.
Double Weakness
If there is one type that both your types are weak to, you will have a "double weakness" to that type and take 156% more dmg. Look for the same colors to the left on both your types.
Gengar
Ghost & Poison
Strong against (+60% dmg)
Grass, Fairy, Ghost & Phychic
Weak to (+60% dmg)
Phychic, Ground, Ghost, Dark
Triple Resistance
If the resistances in common on both your types are one resistance and one immunity, you will have a "triple resistance" and take 76% less dmg. (This is quite uncommon)
Resists
Fairy & Grass (38% less dmg)
Normal (immune) (61% less dmg)
Double Resist Poison & Bug (61% less dmg)
Triple Resist Fighting (76% less dmg)
Zapdos
Flying & Electric
Contradictive effects
As mentioned above, if a pokemon is weak to a specific type on one type, but resist it on the other type, they cancel each other out and deal neutral damage. Immunity counts twice and can cancel out a weakness, and still count as a normal resistance after that.
Strong against (+60% dmg)
Grass, Fighting, Bug, Water& Flying
Weak to (+60% dmg)
Rock & Ice
Summary Dual Type Interpretation
Attacks
Your move type determines if the damage is super effective or not, find your type and follow the right hand side side of the table.
Defence
Find your two types, all your weaknesess is to the left and "adds up" on both types. Cancel out any weaknesses that a resistance or immunity on the other type can compensate for. Add together the resistances underneath both types.
Thats it! If you get the hang of this, you can analyze your pokemons attacking and defensive capabilities against any type or pokemon without looking this up individually for each pokemon. This skill is very important when analyzing the coverage and balance for you team! When you get used to it you can see weakness of two pokemon at the same time by scanning the color coded chart just by "eye-balling" :)
Resists
Ground, Grass, Fighting, Bug, Flying & Steel (38% less dmg)