Here are some great options for beginners (in order of difficulty)
Option 1 - Mt. Takao's Inariyama Trail (dirt trail)
Getting There
Train station 🚉:
Takaosanguchi Station / 高尾山口駅 / https://goo.gl/maps/RZAncG96KWGftEGEA
Travel from Shinjuku? 1 hr
9 am -> 9:43 am via Mt.Takao-Go 1 Rapid-Liner (⚡ Needs Seat reservation; ¥800)
9:01 am -> 9:56 am via Limited Express + Local (Only ¥390)
Overview
Takaosanguchi Station -> Hike 🥾 -> Takaosanguchi Station
Food
No stress, just bring water
Cafe & Combini at the station & on the mountain
Hosoda-ya / 細田屋 - Soba shop on the trail
Summary
Easy, can get a bit busy
Hiking Poles 🔧 = 1/5 not needed ~
Trail Running 👟 = 4/5 Totally doable ~
Super easy to get to from Shinjuku
Option 2 - Mt. Mitake + Rock Garden
Getting There
Train station 🚉:
Mitake Station / 御嶽駅 / https://goo.gl/maps/DMhiun282BQ1sZGS9
Cable Car station 🚠:
Mitakesan Station / 御岳山駅 / https://goo.gl/maps/fUiVTkKLTc9JgUx59
Travel from Shinjuku? 2 hr
7:44 am -> 9:04 am {Mitake Station} + 9:12 am bus -> 9:30 am Cable Car -> 9:36 am | Details
Overview
Mitake Station -> Bus -> Rope way -> Hike 🥾 -> Rope way -> Bus -> Mitake Station
Food
Bring some snacks & water
Grab lunch at Mt. Mitake
駒鳥売店
Summary
Easy & Pretty 🌳
Hiking Poles 🔧 = 2 ~ 3 / 5 ; Nice to have for the Rock Garden trail
Trail Running 👟 = 2/5 ; pretty steep & rocky so not good for running
Simple to get to Mitake station from Shinjuku on express trail (morning)
⚠️ MAKE SURE YOU KNOW THE CABLE CAR TIMETABLES! HAVE PLENTY OF TIME FOR THE CABLE CAR!!!
Option 3 - Mt. Jinba + Strange Horse
Getting There
Train station 🚉:
Travel from Shinjuku? 1 hr
8:50 am -> Takao Station -> 10:06 am via Keio Limited Express + Chuo Local | Details
Overview
Sagamiko Station -> Hike 🥾 -> Jinbakōgenshita Bus Stop -> Sagamiko Station
Spec
Food
Summary
Up hill but simple; Soba at the top 💪
3.5 km (42 min) of walking on road as you head to the bus stop 🥲
Hiking Poles 🔧 = 3.5/5 ; Good to have for the up hill section!
Trail Running 👟 = 2/5 ; pretty steep & lots of roots so not good for running