Place must go:
danshui (go for sunset), jiu feng, yeliu, zoo(got panda), taipei 101.

hot spring
beitou, wulai, yangminshan

Also can consider travel by train to pinxi, sifen.

nite market
shiling, jilong,huaxi, linjiang, shida

shopping
ximending, east area (all the zhongxiao mrt station). all the SOGO in taipei


most of the shop and restaurant accept credit card.
wu fen pu not sure. nvr buy thing there.