From the Super Mario Wiki
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Training Barrels are a type of Barrel found at the very beginning of Donkey Kong 64. They are floating barrels with Cranky Kong's face painted on them and serve as the tutorial of the game. Jumping in them will transport Donkey Kong to a location inside a giant barrel and require him to complete a timed challenge relating to one aspect of the game's controls.
Before giving Donkey Kong his first potion, Cranky Kong will require Donkey to complete his Training Barrels. The four barrels will then spawn in an corner near Donkey Kong's Treehouse, called the Training Area. After jumping in one of them, Squawks will appear and explain the objective of the challenge and the controls required. The four challenges are as follows:
After completing one of the tasks, the corresponding Training Barrel will explode. When all four are completed, Cranky Kong will give the potion for the Simian Slam, allowing Donkey Kong to hit a switch and access the main overworld.