From Tokyo or Shinjuku station to Okutama, use JR. Chuo line, about 2 hours, 1,210YEN. You can begin one of many hiking courses from a few stops before the terminal stop Okutama, for example, get off at Hatonosu station, hike a nice trail which follows to Shiromaru station, thenceforth to Okutama terminal station. If you are interested, try it! I hope you enjoy this course, called "Ohtama Walking Trail".
At the terminal Okutama Station, there are some weekend express direct return trains for Shinjuku Station, which are the afternoon trains marked with purple font in the URL below which links to the official JR timetable. Other black font times go as far as Ohme (unmarked) or further onward to Tachikawa (marked with a simple Chinese character). From either of the latter two stations, you may transfer to more frequent inbound trains.
From Shinjuku, take JR Chuo rapid Line towards Ome direction and transfer at Tachikawa to Okutama. Then take bus to Okutama-ko. Train one way takes about 2 hours and bus one way is about 10-15mins.