There are 20 regions in Italy, including two islands, and every single one is served by Italian trains. The degree of service in each region, however, differs quite a bit.
Travel by train in Tuscany, Lazio, Lombardy, Veneto, Piedmont, and other heavily-touristed and more densely populated regions is easy. Cities of all sizes and many small towns have train stations, and the popular routes between regions have frequent train service. High-speed trains cover most of the major cities in these regions.
In the less-traveled regions—particularly in the southern part of Italy—train travel is still available though a bit less robust. There are high-speed train lines in some of the southern regions of Italy now, including Puglia and Calabria, but service is less frequent than in the north. Many of the smaller towns in the south or the mountainous north do not have train stations or, if they are on the train line, frequent service.
The islands of Sicily and Sardinia are two of the regions in Italy where it's most challenging to get around by train. Visitors in those regions, and in much of the south, are often better served with a rental car.