By train: Trains run directly from Milan Centrale once every two hours during the day, taking about an hour. About €6 each way. Once you are beside the lake, the ride becomes very scenic.
Varenna is very small, and is best enjoyed on foot.
The biggest draw of Varenna is simply to be able to stroll around by the lakeside or through the winding alleyways, enjoy the traditional Italian architecture and take in the serene atmosphere.
The castle of Vezio whence you can admire a beautiful panorama of Lake Como. Also, you can meet the Falconer of the castle, showing birds of prey to tourists.
 Note that the castle appears to be closed in winter, only open between March-October.
Albergo del Sole by the village square offers a good range of Italian food. They also have a range of delicious fish from Lake Como.
There are a few bars around, specially on the lakeside. L'Orso is last one to close (at 2 am).
Take the ferry and explore the villages on the opposite side of the lake: Menaggio, Griante or Bellagio. Go by car or on foot.
Head either north or south by train. Note that there is no ticket office in the station. You are advised to either buy tickets from the travel office near the station, or to buy tickets from the conductor on the train itself.