New Horizons has examined Pluto and will continue to examine at least one more Kuiper Belt object, but it's not aiming to measure the interstellar medium, let alone nearby stars; it's too slow for that.
It didn't go into orbit in the Pluto-Charon system; there's no atmosphere thick enough for aerobreaking, and it would take too much energy to slow down. It made a flyby on 14 July 2015.
Its next stop is the Kuiper Belt object 2014 MU69. After that, it will fly on and ultimately, in the 2040s or so, leave the solar system. It's slower than the Voyager spacecrafts.
