tour grp will bring you to the main tourists attractions but might not be able to see most of the local sights.
Free and easy needs some researching and planning and can see more at your own pace. up to you what is best for you. I do prefer free and easy since i get around there no prob by myself.
Actually if you don't have the time to plan or just don't want to tie yourself down with nitty gritty details like where to stay, ait tics, travel insurance and how to get around, then its better to go with tour agencies, but be prepared for very standard stuffs (touch and go attractions).
Since taiwan ROC is pretty much a mandarin speaking country, personally I would plan my trip. If it's some place where the majority of the population dont understand what i'm saying and I don't have a friend there, then its probably safer to go with a tour agency.
take bus from cks airport to taipei bus station and from there, can take taxi or walk to Xi Men Din. last time while i was perm stuff there, we used to check in Madonna Hotel near Xi Men Din, tell them we are Xin Guang (Starlight) will get discount, dun noe if still exist.
in taipei, can cover Xi Men Din, Hua Xi Jie ye shi(night mkt), Gu Gong, wu lai...