For the most part, Taiwanese people in the bigger cities have studied English since grade school. They can be a bit shy to use it though, so you'll likely need to open your mouth first. But they can surely understand enough to have a conversation. The older people will generally have a lower level of English, but anyone age 45 and under should be conversational.
Shopkeepers and store clerks can speak basic English related to their work after dozens of encounters with Westerners. I've been most impressed by meeting many university students who have a deep understanding of the colloquialisms of English. After all, most of them have studied abroad in the US, Australia, or Europe at some point. My advice is don't be shy, and start chat chatting with Taiwanese wherever you go. You'll be surprised every time.