There is no Cow Level but there is Whimsyshire
As it says on the loading screen and as Blizzard has always said, there is no cow level! However, there is Whimsyshire. Whimsyshire is a zone that can be visited using the Staff of Herding which can be made and further upgraded through the Blacksmith. The Staff of Herding recipe drops off of Iskatu, the boss right before Diablo in Act 4. In order to create it you need five different items and a little bit of gold. The same applies for upgrading the staff, each time you'd like to access Whimsyshire on a higher difficulty you will have to pay a price at the Blacksmith to upgrade the staff. The good thing about this though is when you upgrade your staff you can access all previous difficulties with the upgraded staff.
In order to get to Whimsyshire players will need to join an Act 1 game and take The Old Ruins Waypoint. From here you will want to backtrack away from where Adria's hut is and back towards Tristram. Where you will be going is a little bit difficult to explain in text and much easier in video format so click here to watch a video of my Whimsyshire run and how I got there.
Now the thing you probably would like to know right now is how to get all of the items to access Whimsyshire. Luckily, I've got you covered! Below is a complete list of all the items you will need to create the staff of herding. Also with each of the items I give you a written guide on how to get them as well as a video guide of how to get them. You just can't go wrong!
Quick Note: A "Pony run" is a Whimsyshire run. It is called a pony run because the mobs you are fighting inside are Killer Unicorns! Ahhhhh!
Farming for the Black Mushroom
The Black Mushroom item that you need to make the Staff of Herding comes from Cathedral Level 1 in Act 1. This level will be traversed earlier in the game when you are on your way to save Cain rather than when you're after the Skeleton King. The easiest way to find this Mushroom is by selecting the Reign of the Black King quest and taking the Cathedral Gardens waypoint. From here you will want to enter Leoric's Passage and take the door to level 1.
The Mushroom can be found somewhere randomly on this level but it doesn't always spawn in every game. Since everything in Diablo 3 is randomly generated (or almost everything) it puts the Mushroom's room on that list as well. It won't take long to actually get the room for you though, maybe 3 or 4 resets and it will be all there ripe for your picking!
Farming for Leoric's Shinbone
Leoric's Shinbone is a random spawn in the fireplace of Leoric's Manor, an area towards the end of Act 1. If you are entering the Manor from the courtyard then the room that has the Shinbone is always on the first right. At the fireplace you will find some Burnt Logs that you can click on. Click on these, cross your fingers and hope for the Shinbone. As with many of the other items that you need for this Staff there is a random chance the Shinbone won't be in the Fireplace. If it isn't then will want to leave game and resume which will put you right back at the door to the Manor.
The easiest quest to start from is The Imprisoned Angel, which allows you to port right to Leoric's Manor and check the Fireplace.
How to get Wirt's Bell
Wirt's Bell is the easiest of the items to get as all you need is 100k and access to Caldeum during Act 2. There is an NPC in The Bazaar named Squirt the Peddler who will sell you this bell for the aformentioned price of 100k. When you buy the bell (as if it was to make the purchase more worth it) you also get the achievement "Wirt's Case Scenario" as a bonus.
Farming for the Gibbering Gemstone
The Gibbering Gemstone drops off of a random spawn named Chiltara whom is in the Caverns of Frost Level 2 in The Fields of Slaughter (Act 3). The closest quest to this area is Machines of War (the one where you destroy the ballistae and trebuchet). If you take the Bridge of Korsikk waypoint and head to the northwest you will find a spawn for the cave nearby. However Caverns of Frost shares a spawn with Icefall Caverns which means this may take a few tries to get it to spawn.
So, once you get the cave to spawn and kill Chiltara inside for the Gibbering Gemstone, you're all set for this part!
Farming for the Liquid Rainbow
For the Liquid Rainbow you need to visit the Dahlgur Oasis in Act 2. Here you will find Zaven the Alchemist who will start a quest where you need to save him from phantoms. After doing so he will open the door to the Mysterious Cave for you. Inside the cave you will be searching for the Mysterious Chest. Much like Zaven the Alchemist, this chest will not always be there in every game. So, unlike some of the previous items there are two factors here.
The quickest way to get to Zaven is by selecting the Blood and Sand quest and using the Path to the Oasis waypoint. It is to the southeast from here.