Hatsumōde is the first shrine visit of the new year in Japan. I was at the Buddhist temple of Asakura that time. This is a picture of the temple. Here you see the place was very crowded around. How come the people visit the Buddhist temple instead of Shinto shrine on the new year's day? Does the religion matter here?
Japanese religious piety is ambiguous. I am not sure if those who go to make wishes for the new year really believe in some transcendental deity or something. I often hear what they wished for things like money, love, and their success at exams or jobs. Japanese concept of god, "Kami", is not specifically like the Judeo-Christianity, but described as those concerning the phenomenon of natural emanation, the dwelling in trees, or forces of nature. It is often said that there are eight million gods (八百万の神 Yaoyorozu no Kami) in Japan. Therefore, when Japanese people say they wished for Kami, we can not usually know which Kami did the person was speaking about.
Perhaps, their definition of Kami includes the Buddha, so they visited the Buddhist temple on the new year's day.
A common custom during hatsumōde is to buy an Omikuji, a written oracle. This custom is held both in shrines and temples and exist as the universal custom to communicate with Kami. The omikuji goes into detail, and tells you how you will do in various areas in your life, such and business and love, for that year. Thus, the Kami's message fit the contemporary Japanese people.