u like shopping? go tai zhong night market (feng jia ye shi), yi zhong jie, shop till you drop.
can also buy tai yang bing, pineapple tarts there very nice
You can go abit out of taipei, to jiufen old street, shifen for sky latern, nearby there is ye liu ( the place where u see queens head), jinguashi (well.. nothinig really fancies me here though, just the part where u got to experience gold mining and see the big gold bar, its very near to jiufen like 3mins bus ride only), can go to huang jing po bu nearby as well. this one not bad.
depending if you doing bus trip or cab.
if bus - ye liu > jin gua shi > jiu fen > keelung night market (note there is also some shops on the way to keelung night market, so u can allocate some time here to shop abit)
or
yeliu > jinguashi > jiufen > shifen
*shifen shop close quite early, i went there at about 8pm mostly started to close liao*
if cab - once you get off rueifang train station, got alot of cab there with some package on the places you can pick anyone you like. price is about $100SGD per cab( bring you to about 4 or 5 places i mentioned above)
i like Hua lien alot. More for sight seeing. I travelled with a cab uncle call Mr Lu. He made our trip really interesting

If want PM me, i give you his contact. you can go ask him how much he charge now. i went like 3 years back.. so probably the rates are different now.
i went alishan as well. but hor.. i dont recommend. no pretty girls la.. i know the alishan song. but all of them become auntie liao la... and its so out of the way and quite boring.
unless you are a person who like trees. (personall opinion.. look like bukit timah hill) haha. the yun hai there is beautiful la.. but if you really want to go high up there.. i recommend you to go cingjing farm. so much more interesting.