Where will New Horizons go after sailing past Pluto? The New Horizons Spacecraft is scheduled to whiz by Pluto around 2015, and my understanding is that it is going to do exactly that — whiz by it.
Where will it go after that?  Or what else can it do once that mission is completed?
 A: There are plans for encounters with Kuiper Belt Objects (KBOs) after it passes Pluto.  (Pluto itself is a KBO.)
Quoting the New Horizons Mission Timeline:

Plans for an extended mission include one to two encounters of Kuiper
  Belt Objects, ranging from about 25 to 55 miles (40 to 90 kilometers)
  in diameter. New Horizons would acquire the same data it collected at
  Pluto - where applicable - and follow a timeline similar to the Pluo
  encounter:
  
  
*
  
*Closest Approach - 4 weeks: object observations
  
*Closest Approach + 2 weeks: post-encounter studies
  
*Closest Approach + 2 months: all data returned to Earth
  

The "one to two" is a limitation imposed by the amount of available fuel.
The tricky part (well, one tricky part) is finding KBOs that New Horizons can visit.  
And the project is asking for help from the public to find suitable KBOs.  It turns out that the human eye is better than computers at identifying potential KBOs in photographs.  You can help by visiting http://www.icehunters.org/.
How cool is that?
A: NASA announced on August 28, 2015, the new destination for the New Horizons space craft:

NASA has selected the potential next destination for the New Horizons mission to visit after its historic July 14 flyby of the Pluto system. The destination is a small Kuiper Belt object (KBO) known as 2014 MU69 that orbits nearly a billion miles beyond Pluto.
New Horizons will perform a series of four maneuvers in late October and early November to set its course toward 2014 MU69 – nicknamed “PT1” (for “Potential Target 1”) – which it expects to reach on January 1, 2019.

