To get Milk in Cookie Clicker, you can complete achievements in the game. In doing so, you will be able to acquire milk at the rate of 4% per achievement. Also, whenever you unlock the achievements in order to get milk, you need to remember that you will only be able to do so by unlocking non-Shadow achievements.
The Milk n' Cookies is a craftable Clicker accessory that combines the effects of the Cookie and Glass of Milk.
With more achievements, you unlock new milk flavors. Eventually, you can choose to display any milk that you've currently unlocked. As quoted from the cookie clicker wiki, "After purchasing the Classic dairy selection heavenly upgrade, you may freely choose the flavor of Milk using the "Milk selector" switch.
Milk is a valuable resource which unlocks the Kitten series of upgrades that greatly increase cookie output rate in direct correlation to the amount of Milk the player has. Milk is gained by unlocking normal (non-Shadow) Achievements at a rate of 4% milk per Achievement.