LapLap's advice is certainly more comprehensive than mine but I wanted to chime in from the perspective of someone who has only visited once, albeit on the first weekend of this year.

I arrived at about eleven from Oshiage having taken the Asakusa Line to Asakusabashi to change to the Chuo Main Line for Takao. At Takao, as mentioned by ksandness, I changed for the Keio Line. For me, it was easier to make the change at Takao instead of Shinjuku but YMMV. It looks like I spent three hours on the mountain followed by an additional hour at the onsen that LapLap mentioned. Being on a short trip to Tokyo over a weekend, I wanted to save time and take the funicular up. However, the lines were a little crazy, so I took the ropeway instead, which is accessed behind the funicular station. I had some apprehension about taking the ropeway (slight fear of heights) but there was netting underneath for the whole journey. I also took the ropeway down, having bought a return ticket at the base station but never felt like I missed out on anything by not walking all the way to the top.

As for the mountain itself, I did the typical thing of walking to and then through the Takaosan Yakuo-in temple complex. Being the first weekend of 2018 and combined with the spectacular weather, there were a lot of people there. After passing through Takaosan Yakuo-in, I completed the walk to the top of Mount Takao where I took in the views of Mount Fuji and rested for a bit before heading back down. While passing the funicular station, I recall buying a sausage on a stick from a vendor. After the aforementioned ropeway ride down, I walked through the omotesando and didn't even realise that the Takao 599 Museum was nearby.

The only thing I wanted to add was that while the Keio Takaosan Onsen was indeed a nice experience, it was also the most crowded onsen I've ever been to in Japan. Once again, I realise that I went to Mount Takao on a very busy weekend but I thought it was worth mentioning. As someone with rudimentary Japanese language skills, it was pretty easy to figure the whole thing out (lots of instructions in English, automated ticket buying machines, etc.). They also have television monitors in the area by the restaurant that show the train departure times so that you can time your departure from the onsen properly.

Hope no one minds pictures

JR Takao Station

Hikawa Jinja, located next to the station

Kiyotake funicular station

Sanroku ropeway station at the foot of Mount Takao

Going up

Mid mountain view from near the funicular station

Walking toward Yakuo-in

Shops at Yakuo-in. Looked like they were mainly for offerings (or something else of that ilk)

Izuna Gongen-do

Mountain top

Mount Fuji in the distance

More people

Passing through Yakuo-in again on the way down. This is the shoro (bell tower)

Lines at Sanjo ropeway station to head back down

Going down

Keio Takaosanguchi Station