If planning a trip to Taiwan, you can estimate the time & cost of train travel by going to the Taiwan railway website. They also have a mobile app for it. I've estimated that I'll use ~2k ntd for the long distance trains. You'll pay a bit more if you use the high speed rail (and get to your destination faster).
Today, I'm staying at a hostel but I have my own room. In Taipei, I stayed in a hotel. I try to mix it up, depending on my budget & needs.
In Taipei, I was toured by a Taiwanese friend (but I didn't stay at his place). Since he was a local & spoke the language, it was very nice. We went to Tamsui, an area north of Taipei. It's near the river and has lots of small shops along the riverside. Kinda like the riverside atmosphere in Phnom.Penh or Vientiane. Since it was a weekend, there were many families, teens & seniors walking about, and it was good to just sit & people-watch.
Sunday, we went to the city of Keelung, then Yeliou Geopark. It's a geological park with lots of rock formations by the coast. Entrance fee was 80 ntd. Since it was a weekend, there were lots of ppl & you couldn't get a good pic of the rocks w/o pesky ppl. Lol. All in all, it was okay. I didn't get to have my photo taken with the "queen's head" rock, since there were lots of ppl queing up for it, and I didn't want to wait. Anyway, I don't post my pics on FB, IG or twitter, so no need for that iconic photo. Heh.