Bumble Bee Mum's blog on a family trip with kids to Taiwan, by rail. This covers Day 1 a relaxing time in Taipei,  Day 2 Taipei to Kaohsiung (E-Da Theme Park), Day 3 day trip to Cingjing (Nantou) then back to Taichung. Day 4 Taichung to Taipei. For travelers seeking a fun family activity, this is the itinerary to use, which is not too packed apart from day 3 making a day trip to Cingjing. You may replace the night market in either Taipei or Taichung to visit other places.

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Total trip length: 5D

Places of interest: Taipei - Hello Kitty Cafe, Shilin Night Market, Kaohsiung - E-Da Theme Park, Cingjing, Taichung - Fengjia Night Market

I picked this itinerary from Janiscooking.net because this is a pretty standard template for travelers and transportation can be fairly easily arranged to go to most of the tourist destinations.

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(selected places of interest featured)

Day 1: Taichung - Lavender Cottage (Elin Chow has a detailed post on Lavendar Cottage), Rainbow Military Dependents' Village

Day 2: (Road trip) Taichung to Miaoli - Flying Cow Ranch

Day 3: Taipei - Taipei 101, Shilin Night Market, Lungshan Temple

Day 4: (Road trip) Yehliu Geopark, Jiufen Old Street, Golden Waterfall and Shifen Old Street/Waterfall

Day 5: Return

Most tourists do not include traveling to Hualien in an itinerary under 5 days. However, I think this is worth trying and have adapted from this Hualien to Taipei 7 days itinerary.

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Day 1: Airport to Hualien (by High Speed Rail HSR)

Day 2: Hualien Taroko Gorge day trip

Day 3: Hualien to Taipei

Day 4: Taipei Jiufen, Shifen day trip

Day 5: Back to home

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Places of interest: substitute the jiufen, shifen raod trip by taking a train through the Pingxi line.