Claps and bows in silence
Every New Year's holiday, I go to Tsurugaoka Hachimangu in Kamakura. There, you first throw money, bow twice, clap your hands twice, pray and then bow once (you can follow this process at any shrine). It may sound like a complicated process, but all you have to do is follow the person before you (there are instructions in Japanese on the process of praying). The beauty of the praying process in Japan is the silence. Everyone is respectful for others and prays quietly and quickly, so that there is enough space for other people to pray. If praying during the first few days of January, remember that you would have to pray quickly - there are many people who wants to do "Hatsumoude" or visit the shrine on the first day of the year.
Watch this youtube video for a more detailed and visual explanation: http://www.youtube.com/watch?v=bWAPCxNo-PU.
Good luck and remember to enjoy the experience!