Zante Town, also known as Zakynthos Town, does not really provide beaches catering for tourists.
There are a few places to swim around the town, but they are not nearly as well developed as the resort beaches.
The main beach front promenade in Zante Town, which is used by many of the locals during the summer for swimming, can be found along the Krioneri beach front. The shoreline is pebbly and rocky and the water is difficult to enter because of stones, so you need swimming shoes. Sun beds are not usually available on the Krioneri beach in Zakynthos Town.
We recommend that during your holiday in Zante, you swim in the beaches around the island resorts of Zakynthos, and save your time in Zakynthos Town for shopping, exploring the sites, and enjoying the many charming tavernas and cultural activities.