(2-part post):
Let's start with an important link. All European rail timetables are incorporated into a single site (in English) courtesy of the German state railway, Deutsche Bahn. Plan all your travel here:
http://reiseauskunft.bahn.de/bin/query.exe/en
Going south from Milan, Genoa is about 75 miles away. There are direct rail services, with the journey taking about 1.5 to 2 hours. Genoa is often thought of as an industrial city but it's got a wonderful historic heart and some great shopping and restaurants. You can also take the cheap, frequent rail services along the coast to explore the Italian Riviera and delightful towns like Rapallo.
Further south, or south-east, there's Bologna (direct trains, about an hour and three quarters), Pisa (direct or change at Genoa, typically around 4 hours) and Rome (direct, around 4 to 5 hours).
To the east, you've got Venice (direct trains, about two and three quarter hours), Trieste (direct or change at Venice, typically around 5 hours), Ljublana (Slovenia, typically 3 changes and around 10 hours) or Zagreb (Croatia, typically 2 or 3 changes, around 10 hours).