When you’re in Taipei, head to Jinshan, Baishawan or Longmen beaches. You can stay in cabins at Jinshan and Longmen. From Longmen you can do the Caoling Historic trail which gives fantastic views of the coast and also grassy highlands. Or bike though the very scenic Taoyuan Valley and see terraced rice fields and water buffalo. You might even think about taking a boat to volcanic Turtle Island or just a cruise along the coast.
Take a train to Ruili and then catch the Pingxi line to Shifen to view the Shifen Waterfall. Then head back to Ruili and take a taxi to Juifen and Jingguashi for quaint hillside teahouses and a look at an old gold mining town.
Keelung had a number of interesting old forts and a cool cave, and the trip to Keelung island is worth taking. There’s also the famous night market if you want to sample local snack foods. Nearby Yehliu is fun for the bizarre rock formations.
Do you want to go to a teahouse in the mountains? I know a couple excellent ones in the southern part of Taipei.
Try to catch a show of Taipei Eye, revue theatre for traditional performing arts.
Of course don’t miss the National Palace Museum. I think there is an Egyptian exhibit at the History Museum now. There is also that body worlds show where real corpses are displayed.
Take your girlriend to the shop in Shimending (a neighborhood in taipei) where they still make traditional chinese slippers by hand. She’ll love them. Also in the same area is the old Red Theatre where they have traditional performaces every day.
For temples go to Longmen and Baoan.
Climb Kuanyinshan across the river from Danshui to see a perfect sunset from the platform atop the volcanoe. Also the boardwalk at Bali (the town across from Danshui, under Guanyinshan) is very romantic at night, especially heading north. You can stand on a pier jetting out into the mouth of the river as it empties into the strait of Taiwan. Guanyinshan is right behind you and across the wide river the mountainous landscape of Yamingshan.
There is also a fun waterworld right there and an interesting aboriginal museum.
What kind of stuff do you want to see? Museum? Nature?