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.
There are a few bars around, specially on the lakeside. L'Orso is last one to close (at 2 am).