After spawning on the "Goat Ville" map, look towards the cornfields. A dirt track is to the left and the goat ring is to the right. It is a red ring with people cheering around it. Step in the ring and a goat will be summoned each round. Push them out any way possible. Do not allow your opponent to headbutt you. If you cannot headbutt or kick them out, knock them down and lick them to drag them near the edge of the ring. Be careful not to walk out of the perimiter yourself. Each time you win a round, your goat gets slightly larger. Defeat four goats to earn this achievement.
Easy "Angel Goat" achievement
Start the game and allow it to idle for five minutes in order to avoid causing any property damage or bodily injury.
Easy "Around the World on 5 Trampolines" achievement
Use the Anti-gravity Mutator to make it much easier to direct yourself when on the trampolines.
Easy "Boulder of Death" achievement
Carefully look in the general direction of the Pentagram to see a boulder on top of a hill. Roll it to earn the achievement.
Slender Goat
Although not an official unlockable character, it is possible to turn your goat into a Slender Man variant. Find all trophies in Goatville to unlock Space Goat. Defeat all the goats in the Fighting Arena in Goatville to unlock Ripped Goat. Then, select "Custom Game" at the main menu. Select Space Goat and Ripped Goat, then choose any map.
Teenage Mutant Ninja Turtles reference
In the Goat City Bay level, find the sewer entrance pipe in the water between the roller coaster carnival and the Put-In Hotel. It is next to the bridge. Enter the tunnel and follow it to find the Teenage Mutant Ninja Turtles.
Various Google Play Achievements
Achievement
How to unlock
Alexander Goatstafsson (9,000 XP)
Win the title bout!
Cardio Kills your Gains (9,000 XP)
Drag a person into the treadmill with your tongue.