How to explore and discover a new city for free? I'd say, join a free walking tour.
During my first two days in Taipei, I joined Like It Formosa's two free walking tours, and perhaps that belongs now to the best decisions I ever made in my travels.
Like It Formosa is an independent organization which promotes what they call "cultural tourism". It is a tip-based downtown tours by young local tour guides. See Taipei in a local's perspective—that is what they promised to do. And they did, I did. We visited historical landmarks with so much rich history, discovered hidden corners that only locals know, and they did it in a very informative way.