This is a rough itinerary. You can check it against google map of Taipei for all the locations.
1st Day:
From Taoyuan Airport, take Kuo Kuang bus #1819 costing NT$125 per pax to hotel near Taipei Main station and from arrival at 3pm, Buy Easy Card for buses, mrt an train rides at mrt platform at basement of Taipei Main station.
plan to walk towards Taiwan Museum near NTU Hospital mrt station,
take mrt to NTU near Gongguang mrt station -http://hsiuchi.prince.com.tw/eng/tour.php ,
mrt to Ximen mrt station – 3 stations for Ximending
2nd Day:
From Taipei Main station, take mrt to Bitan suspension bridge – southwest of Xindian mrt station – 11 stations.
Take mrt to NTU Royal Palm at Siyuan street near to Gongguan mrt station – 6 stations,
take mrt to Chiang Kai shek station – 3 stations,
take mrt to Ximen station – 2 stations and change mrt to Longshan Temple – 1 station,
walk 5 minutes to Huaxi Market
3rd Day:
From Taipei main station, take mrt to Zhongxiao Fuxing mrt station – 3 stations for Sogo Dept Store,
mrt to National Sun Yat – 2 stations,
take bus or walk to Taipei 101,
From Taipei 101 at Shife Road, take bus #281 to Wu Fen Pu
or at Songzhi Road, take bus #28 to Raohe Market,
4th day:
From Taipei Main station, take TRA train to Hualien station – 2 hrs 10 minutes costing NT$440 per pax. Hire a taxi to Taroko George or buy 1 day pass at Hualien bus station on left side of train station.
You can also take Hualien Buses to Taroko Gorge. 1 day ticket cost NT$250 per pax from Hualien train station so you can get up and go down whenever you like. If you do not buy 1 day pass, you have to pay with exact cash whenever you go up bus. Driver will not refund change if you pay more.
Buses leave from Hualien station at : 0630, 0750, 0840, 0900, 0940, 1020, 1050, 1130, 1230, 1350, 1400, 1500 hrs.
Buses leave Tianxiang from: 0910, 0940, 1040, 1200, 1310, 1400, 1430, 1600, 1640, 1700, 1800 hrs.
1900 hrs – take TRA train from Hualien station back to Taipei Main station about 2 hr 10 minutes costing NT$440 per pax.
5th day:
take mrt to Zhongxiao Fuxing Road, exit 1 to take bus #1062 to Jiufen costing NT$102 per pax about 60 minutes.
1600 hrs - After Jiufen, take bus #788 down to Riufang station – 15 minutes costing NT$30 per pax.
Go into Riufang station to take train to Shifen station – 4 stations – 26 minutes costing NT$20 per pax,
explore old street and release sky lantern.
Take train back to Riufang station. Last train depart Shifen at 1030 hrs.
At Riufang station, change train to Taipei Main station about 35 minutes.
6th:
From Taipei Main station, take mrt to Tamsui/Danshui station.
Take bus R26 or #857 to Fisherman wharf.
From Danshui/Tamsui mrt station, take mrt to Beitou mrt station
take bus 230 to Beitou Hotsprings, Hot spring museum, Hell valley.
After Beitou, take bus 230 back to Beitou station to take mrt to Jiantan mrt station,
exit 1 to walk to Shilin Night market.
After Shilin NM, take mrt back to Taipei Main station.
7th day:
From Taipei West bus station A Terminal, take Kuo Kuang bus #1815 to Yehliu bus stop,
walk 15 minutes to Yehliu geo park.
After Yehliu, walk back to same bus stop to take bus #790 or 1815 to Jinshan.
Take bus 1815 back to Taipei Main station.
8th:
From Taipei Main station, take mrt to Shilin mrt station - 6 station.
Exit 1 to take bus R30, 255 or 304 to National Palace Museum.
Take bus R30, 255 or 304 back to Shilin mrt station
and change mrt to Taipei Main station.
How cold/hot will it be in October? About 19 – 27 degree C. accuweather.com/en/…
