Let's talk about trail choices. Three trails lead from the base to the top of the mountain.
Trail 6 heads up the left-hand side of the mountain, following a stream gully most of the way to the top. It's more of a nature trail than Trail 1, heading through a forest and offering lots of beautiful natural sights. At one point, near the top of the mountain, you follow a stone path right up the stream (which flows at a trickle in that area).
The third trail that goes from the base to the top is called the Inariyama Trail. It is just a little further "left" and follows one of the ridges that runs alongside Trail 6 and the stream. This path is especially worth hiking in early summer when hydrangeas and lilies are in bloom.
Trail 2's loop goes around the monkey park and garden, and takes about 30 minutes.
Trail 3 branches off to the left, following the ridge between Trail 6 and Trail 1. It faces south, so it has plants that grow in warmer climates; the trail also has three bridges and a giant cedar tree.
Trail 4, which heads up the northern side of the mountain, has a suspension bridge and a lot of shade, though it includes a very steep section near the top.
Trail 5 is another loop trail around the top of the mountain.
A final trail, found at the top of the mountain, is called the Mount Takao/Mount Jimba Trail. It's an 18.5 km trail that takes up to 7 hours to hike one way. It's possible to access Mt. Jimba or a couple other points along the trail with bus service, though I don't know anything about it.
Mount Takao is easily accessed via the Keio Line - a rapid train goes directly to Takao-san-guchi Station every 20-30 minutes or so, with local trains running about every 10 minutes otherwise. It's possible to hike from JR Takao Station (a much further, less-popular option) as well. "Admission" to the mountain is completely free, and the Keio Line train fare from Shinjuku is under 400 yen each way.
If you want to use the cable car or ski-style lift, it's 470 yen one way or 900 yen round trip. The cable car runs from 8:00 every morning, ending some time between 17:00 and 18:30 depending on time of year and day of the week. I'm not sure about the lift times.