The left part of the island is more suitable for holidays with children or parents. There aren't many young people, mostly either schoolchildren aged 16-19 or their parents.
There are two big cities: Polis and Paphos (the latter is entirely a UNESCO World Heritage site). You can find all sorts of ancient things, enjoy mosaics or the remains of once-residential buildings.

Interesting fact: in Polis, for about 200-250 euros, you can rent a boat without a captain (!) for 3 hours for 5-7 people and cruise along the coast. No boat license is required. The coast is rocky and very beautiful; I sat on the grass and watched a man who had sailed on a yacht to admire these beauties.

By the way, we rented a two-story villa (I would have called it a house, but because of the small courtyard, we agreed to call it a villa) with a view of the Mediterranean Sea, two bedrooms, and an outdoor barbecue area for only 100 euros per day! For comparison, spending a night somewhere in the simplest hotel in Paris would cost about the same.