Japanese people eat special dishes called osechi ryori during shogatsu. They are packed in a Jubako box, which has several layers. Each dish has a particular meaning. For example, prawns for long life, herring roe for fertility, and so on. It is also traditional to eat mochi (rice cake) dishes during New Year's holidays. Zouni (rice cake soup) is the most popular mochi dish. The ingredients vary depending on regions and families.
There is a custom of giving money to children during New Year's holidays in Japan. It's called otoshidama. If you are going to family gatherings, it's good to prepare some money in small envelopes.
It is traditional for Japanese people to visit a shrine or a temple during New Year's holidays. People pray for safety, health, good fortune, and so on. The first visit to a temple or shrine in a year is called hatsumoude. Many well-known temples and shrines are extremely crowded. Some temples and shrines expect a couple million visitors during New Year's holidays each year.
Most of businesses in Japan are typically closed around the 29th or 30th of December to the 3rd or 4th of January, depending on the kind of business and day of week. In recent years, many restaurants, convenience stores, supermarkets, and department stores are open even during New Year's holidays. Many department stores hold New Year's special sales, so it might be worth it to check the prices out if you have shopping needs.