If you want to get out in the hills you need to either own a scooter, rent a car, or make friends with cars.
Don't do the bus thing as it can be crippling. Good for going between cities, not for getting into nature.
Views are nice from Peng Chau. They can also be nice from Yangmingshan.
Food is much better and more affordable in Taiwan.
It's easier to make friends in Taiwan.
Yehliu has decent beaches.
Taipei is different from the rest of Taiwan. Basically you'll find everything you need.