Taipei, Hualien, Kenting - what to do/hot springs/trains

It's not far by train from Taipei to Hualien, less than 3 hours I would say. There are hot springs in the Taroko Gorge area, I can't be specific sorry. The railway doesn't actually run to Kenting so you will probably have to take a bus all or part of the way.

WeiLong in Taiwan

For hot springs in Taipei, Beitou seems to be the most popular. If you are willing to do a day trip I think Wulai is even better regarded.
You could spend 2 days wandering around Taroko. If you want to only spend one and have to be in Hualian for the second, there are a few sights in and around Hualian that you could see. Perhaps your time would be better spent at the Chihben hot springs in the southeast, or at one of the sights in the northeast.

The best hot springs in Taipei are on Yanming Mountain. You can take a bus to the hot springs area (forget the number). Xin Peitou is ok but lacks the outdoor natural beauty of Yanmingshan. Wulai is even more beautiful, but a bit further away. You can take a bus from Gongguan MRT (again I forget the number)

Taroko has hot springs in the park not far from the tourist village. There is a catholic hostel there that is clean and cheap. The most ideal way to explore Taroko is by renting a scooter in Hualian and driving. Taroko is one of the most beautiful spots on earth, BTW.

Chihben (sp) is a 15 minutes drive south of Taidong in the south east. Water quality-wise, these are my favorite hot springs. Continuing further south along the coastal highway, you can drive (by scooter) to Kending. As someone mentioned before, the train doesn't go there.

You can also take a bus to Wulai from Xindian MRT. Xindian's bus station is all in one place, unlike Gongguan's which is up and down the road about 5 mins walking in either direction...so if you don't know just which bus you want, which way it is going, and where it is in that strip, you're in for an annoying time. The bus from Gongguan also stops at Xindian - as do one or two other bus routes that begin in Xindian - so I recommend just getting on at Xindian instead.

I agree that Xinbeitou is not as beautiful nor as good as Yangmingshan, though it is a lot more convenient. Bus 260 starts at Taipei Main and goes through Ximen (Zhonghua Road/Chengdu Road intersection) and goes to Yangmingshan, but I am not sure what, if any, hot springs it stops at on its route. There are other buses from Jiantan and Shilin but you'd have to ask locally for those numbers, I don't know them.

There is also a good hotspring town between Hualien and Taidong...I forget the name but it's in LP.

Taroko is truly beautiful, though honestly I felt areas at higher altitudes were even prettier. If you do rent a scooter consider a two day trip - one to do lots of hikes in Taroko, staying in Tianxiang instead of Hualien that night, and the other to drive up to the far higher mountains beyond - Dayuling, possibly Hehuanshan if you are a good mountain driver, Cingjing and other scenic spots are within a second day trip setting out from Tianxiang...then return to Hualien that evening - though that'd be a loooooong drive back.

I've never been to Kending so I can't help there, sorry!

Taipei: as the other said
Hualien: try An-tong you can take the train there
Kenting: Si Chong Xi is not far away from Kenting and it's a famous hot spring spot. The Japan Emperor's brother had been there when during his honey moon.

Thanks so much for the replies guys. really helpful.

Unfortunately I cant ride a scooter (or even a bicycle for that matter) so not an option for getting around. Depending on how much it is we may consider hiring a taxi for the day to go around Taroko, unfortunately doing that for 2 days would prove pretty expensive so likely we will only do one day in Taroko.