Paphos has a daily fruit and vegetable market which is in the old part of town. In the old town you will also find a wide range of market stalls selling souvenirs, glassware, leather goods, watches and footwear among other items.
Just head up the hill from the seafront and turn left down Makarios III avenue. The markets are at the end of this road.
You will also find craft fairs at various times of the year and a good place to find interesting goods made by local people is at Enplo - this is a stone building at the end of Paphos harbour which exhibitions just about every week of the year including jewellery, stained glass, sculpture, paintings and much more.
If you grab a copy of one of the monthly magazines such as Grapevine or Cyprus Living, you will also find details of any up and coming fairs and market stalls.