The fastest way is to take the Odakyu Romancecar train from Shinjuku Station directly to Gotemba (95 min). This train is not covered by the Japan Rail Pass and runs only on weekends.
The route covered by Japan Rail Pass is:
From Tokyo Station or Shinagawa Station take shinkansen to Mishima Station, change to JR Tokaido Line to Numazu Station, change to JR Gotemba Line to Gotemba Station. Total travel time with transfers 117 minutes.