Excellent suggestion. Getting your hair cut here IS a unique experience. Go to a full service salon where they'll wash, cut, and style your hair, then give you a head/neck/shoulder massage. I don't know what the prices are like up in Taipei, but down south, a full service wash/cut/massage will run you about NT$300-600.

They also have the budget places where you walk in and feed a NT$100 note into a machine, and it spits out a numbered ticket. When your number comes up, you give the ticket to your barber, and they give you a basic haircut (no wash/massage).

30 years ago, you had to be careful about going into barbershops, since many of them back then offered...um..."other" services. You learned quickly to avoid the ones with flashy lights and darkened windows. But those types of places are long gone.