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. Don't worry — it's quite frequent. 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.
Once you arrive at Chino Station, head to the Alpico Transportation Information Center (see it on
Google Maps) and take the Shirakaba-ko–Kurumayama Line bus bound for Kurumayama Kogen. This local bus will take you to the start of the hike in about an hour. Be sure to check the bus schedule before your trip. There is no English version of the website, but you can use Google Translate to view the timetable
here (look for the Lake Shirakaba / Kurumayama Kogen Line — 白樺湖・車山高原線).