The old town is a good place to hunt for a bargain. It's a bit of a maze of narrow, intertwining streets, but locally produced goods are available in lots of the shops. Basket ware, pottery, embroidery, copperware, lace and honey are all good buys.
Another good place to buy gifts and souvenirs is the government-run Cyprus Handicraft Service, where you can watch the craftsmen work before deciding what to buy.
Away from the beach, the Paphos Aquarium is fun, or you could visit nearby Polis, which sits on the edge of the Akamas peninsula nature reserve, home to dense forest, lots of wildlife and several nature trails.
You'll need a hire car, but a trip up into the Troodos Mountains is well worth the effort. Enjoy the cool pine and cedar forests and the charming villages and take a jumper just in case, as parts of the region are at around 6,500 feet.