Power berries can be earned in the following ways:
1. Get a horse to five or six hearts worth of affection then enter it in the Spring or Fall horse race. Win that race and you receive a Power Berry.
2. Win the swimming competition for the first time in the Beach festival and you are awarded with a Power Berry.
3. Make sure you ship a pumpkin in the week before the Pumpkin festival. On the day of festivities you receive a Power Berry from the Mayor.
4. Obtain 30 Musical Notes and visit the Harvest Goddess Spring to claim your reward. Make sure you have seen every event for every five notes before as well, or you will need to keep entering until you catch up.
5. Reach the 100th floor of the Moonlight Mine to receive a Power Berry from Tim.