When you enter a Shinto shrine, there are a number of things you can do. If you are like me, you will simply enjoy the serenity. Others will admire the architecture or the surrounding nature. Of course, it is a shrine after all, so one can offer prayers, be cleansed by a priest, get married, or partake in any number of other religious activities. One of the most fun and interesting traditions at a shrine are ema, or small wooden tablets where prayers (祈) or thanks (感謝) can be offered.
Whether you practice Shinto or not, you are welcome to purchase the small wooden tablets at one of the open windows near the honden, or main hall of the shrine. They can usually be purchased for anywhere between JPY500-1000 (the proceeds support Shrines and their employees). From there, you offer either your wish or your thanks by writing it on the tablet, and then you have the choice either to hang it in the shrine or take it home with you (in today's age of cell phone cameras, I recommend hanging it up and taking the photo as the keepsake). You will not have any trouble finding where to hang it--look for the rack with all of the others!