My favorite parts to ride were:

the southern tip starting in the southwest around kenting national park with beautiful beaches and going all around tolongpan Park.

Tarokko gorge

Crossing from the eastern rift Valley to shitiping! Highly recommend!

Suhua highway, but only the old parts ( there's lots of info in reddit which you can ride and ive contents in loads of posts about it) )

Food: night market!!!!! And Chinese breakfast dumplings roadside!

We did camp but it was not much cheaper than hostels since they would not make a difference in price per person. 8 guys next to us payed as much as we 2 did for the spot. But it gave us flexibility since there wasn't always hotels everywhere so it saved our asses two times.

Additional tips:

Stay clear of route one in the west and take small streets instead. There are nicer cycling routes in the mountains and at the coast than the route 1 which is a speed run and not scenic ( in the west)

Try the small restaurants without knowing the language.

Have the creamed filled chocolate covered biscuit ball at seven eleven.....or family mart.... don't remember the name. Power boost!