2 Days in Taipei: The Efficient Itinerary

Traveller Tips

Buy an EasyCard at the airport MRT machine for about $3.20 deposit and top up $10; it covers every MRT, bus, and convenience store on this plan.
Two days is enough for the core landmarks and two night markets but not day trips; save Jiufen for a third day.
MRT runs until roughly midnight, so both night-market dinners get you home by train without a taxi.

Day by day

Day 1: Old town temples to the tower

Longshan Temple at Longshan Temple MRT, 1 hr
Walk the Bopiliao Historic Block lanes nearby, 45 min
Beef noodle soup lunch in the Wanhua lanes, 1 hr
MRT to Chiang Kai-shek Memorial Hall for the honor-guard change on the hour, 1.5 hr
Rest valve: tea and a sit-down at a Yongkang Street tea house near Dongmen MRT, 1 hr
MRT to Taipei 101, ride up at 4:30 PM for daylight plus city lights, 1.5 hr
Dinner and dessert at Raohe Street Night Market by Songshan MRT, 1.5 hr
Total: about 5 km walking; 4 MRT segments

Climb Elephant Mountain from Xiangshan MRT for the Taipei 101 panorama, 1.5 hr
Scallion pancake and bubble tea breakfast at a street cart, 30 min
MRT to Maokong Gondola at Taipei Zoo station, ride the standard cabin up, 1.5 hr
Tea-plantation walk and viewpoint at the top of Maokong, 1.5 hr
Rest valve: hillside tea house with a city view at Maokong, 1 hr
MRT to Ximending to browse the shopping lanes, 1.5 hr
Mango shaved ice and dinner at Ningxia Night Market near Zhongshan MRT, 1.5 hr
Total: about 6 km walking; 1 gondola ride and 4 MRT segments

Traveller Tips

Climb past the first crowded Elephant Mountain platform to the higher rock outcrops; the view is better and the crowd thins, costing nothing.
Take the standard Maokong Gondola cabin, not the glass-floor Crystal Cabin; same fare of about $3.20 one way but a far shorter queue.
Eat dinner at the night markets rather than sit-down restaurants; two full plates plus dessert run about $8 per person versus $20 in a restaurant.