Hualien: Take the tilting train called Taroko from Taipei main station to Hualien. From there you can take a taxi thru 20 kilometers of the Gorge to Tienshiang (Tianxiang). And you can stay in Formosa Regent hotel. A nice hotel. Or a few other nice ones that others can recommend.
You can either return the same way or continue down the East Coast to Taitung by train. It’s about 4 hours, I believe. And I would recommend a stay at the Rolyal Chihpen hotel in Taitung. I enjoyed it there when I was there.
Then you can take another train further south towards the Kending area. Lots of local buses or taxis to take to Kending. I personally like the Caesar’s Park hotel or the Howard Plaza next door. Both have access to this great public beach right there and swimming pools on the grounds.
You can get to Kending on the West Coast from Taipei by High-Speed Train to Kaohsiung and bus, or regular trains or buses or plane. And then there are buses to the Kending area every 20 mins or so 24 hours a day .
To Sun Moon Lake. Spend a night in Taichung (accessible from Taipei by High-Speed Rail or regular train, or buses) and then there are buses to Sun Moon Lake. Or you hire a cab. I dont know how much a cab ride would cost for the 70 kilometers or so (if I remember correctly).
Sun Moon Lake has some nice enough places to stay.
I will let others pitch in here
enjoy ur trip
p.s. those hotels are on the pricey side. There are many other hotels in HUALIEN that are nice and not as expensive, down in the Kending area as well and of course many hotels in TAichung.
EDIT: ok , lets talk inexpensive methods here:
I dont think there is a bus to Hualian anymore is there? But that would be cheaper then a train and much more scenic. And there are many different kinds of trains to take. The more stops they make , the less expensive. And there are local buses that run from Hualien city to and from Tienshiang (20 kilometers into the gorge).
Taichung has buses to Sun Moon Lake, every hour or two.
Kaohsiung has buses 24/7 to Taipei for about 350nt to 450nt I believe. The planes run around 1500 or so, similar to the HSR .Please feel free to chime in with specifics anyone. And there are different classes of TRA trains. Once again, the more stops , the slower and therefore the lesser the charge.
Plenty of nice and inexpensive hotels to stay at anywhere in Taiwan. Except for Taitung. Which is a town I would give a miss entirely unless I was staying at the (expensive) ROyal Chihpen. Correct me if I am wrong but TAitung (the town) really seems to have no attractions??