When my wife and I first started travelling, every dollar counted. We would avoid buying water as best we could and would pack our days worth of water wherever we went. However, when visiting Italy, clean drinking water was surprisingly easy to come by. Perhaps it shouldn't be surprising for a country famed for their aqueducts but water availability has become a part of the Italian culture. We were amazed to find fountains (I don't mean fountains like Trevi Fountain), taps, and even a machine by the coliseum which dispenses free mineral water. Just bring a reusable water bottle and you will be good to go.
The water taste varied and in our opinion was the best in Cinque Terre. While hiking the trails at Cinque Terre, water is generally available at small taps as you enter or leave each town. We found them pretty obvious so hopefully you do not have any issues finding them. This is a lifesaver in the summer when climbing up and down in the heat. Don't worry about saving water for the whole trip as you will have a chance to fill up as you go.
When travelling on a budget it is great to not have to worry about yet another expense and also to stay hydrated. So don't worry about buying too much water when in Italy! I hope that someone finds this tip helpful!
The Amalfi Coast has much to offer, such as small shops selling local products like the renowned limoncello. It contains wonderful historic sites such as the Cathedral of Amalfi, and, most importantly, the dramatic coastal scenery that surrounds the area: cliffs and stunning waterscape.