Getting to Kurumayama Highlands is not that hard even if its your first trip to Japan.
First, go to Shinjuku Station and take a limited express Azusa train to Chino Station.
You can read more about this train and online booking service
here. Thought the trip takes around 2 hours, you can watch the varying mountain landscapes of Mt. Fuji, Yatsugatake, and the Akaishi Mountains (also known as the Southern Alps) pass outside their windows on your way.
After you arrive to Chino Station go to Alpico Transportation Chino Station Information Center (check Google map
here), and take Shirakaba-ko~Kurumayama Line- Kurumayama Kogen bus. It is a local bus that will take you to your start hiking point in 1 hours. Be sure to check the bus schedule before your trip, there is no English page, but you can use Google translation to scheck the schedule
here (you need Lake Shirakaba/Kurumayama Kogen Line/白樺湖・車山高原線)