All these three dishes are composed of either vegetarian stock or chicken broth, plus some other ingredients.
Zuppa is made of veggies simmered for hours until the texture is thick. It is accompanied by slices of toasted bread (crostini) that are dipped into the soup to absorb its flavor.
Minestra is made of broth and includes either rice or small-size pasta. It is widely spread in northern Italy tradition, especially in Lombardy and Veneto, where, according to tradition, they are served in the evening.
Minestrone is a type of minestra having more or less broth but with a great quantity of veggies and legumes. You can add rice or pasta if you like.