The ascent passes through ten stations, with the first located at the foot of the mountain and the tenth at the summit. There is, however, no need to climb all the way.
Sealed roads go as far as the fifth station, predictably around halfway up the mountain, around 2,300 metres above sea level. There are four, with four routes to the summit: the Lake Kawaguchi, Subashiri, Gotemba, and Fujinomiya routes and four more from the foot of the mountain to the fifth stations (Shojiko, Yoshida, Suyama, and Murayama).
Assuming you're not interested in climbing all the way, the most popular fifth station takes you on to the Kawaguchiko route.
It's not the closest to the summit (that is on Fujinomiya), but it has a larger car park, is most easily accessed from Tokyo and has the most mountain huts where climbers can rest or stay.
Depending on your starting point, the ascent from the fifth station takes between three and eight hours with from two to five hours needed for the descent. It takes about an hour to get around the crater and its eight peaks.
Assuming you're not interested in the climb and possible issues with altitude sickness, popular Fuji-viewing locations include the Fuji Five Lake (Fujigoko) region on the northern side of the mountain.
Slightly further away, Lake Ashi and the Hakone region also provide highly rated views of the mountain. In favourable conditions it can be seen from Yokohama, Tokyo, and as far away as Chiba, Saitama, Tochigi and Lake Hamana.
It can also be seen from trains travelling between Tokyo and Nagoya (and thence Osaka and Kyoto) with the best view around ShinFuji Station. But there's no guarantee.
Clouds and poor visibility often obscure the mountain, even from the relatively close shinkansen line (speaking from experience). Visibility is said to be better during the cooler seasons than in summer, and early morning and late evening are reckoned better prospects than the middle of the day.