Binghamton is a city in the eastern most extent of the Southern Tier of New York and the lower right-hand corner of the Western Tier of New York. At the crossroads of the Southern Tier Expressway (NY Route 17) and Interstates 81 and 88, it serves as one of the principal gateways to western New York.
It is very easy to get from Rochester to Binghamton. Get on I-390 and head south. When that road ends, go east on I-86/NY 17, and, sooner or later, you'll get there. As for what to do when you get there:
See the Binghamton Mets (baseball) or Binghamton Senators (hockey) play
Try a Spiedie
In August go to the SpiedieFest and Balloon Rally
See their Rochester page or link to it using: