1. They close the entire country, for about three days, starting on January 1. There are no stores, restaurants, or government offices open. The trains do run, so someone is working. And there is mail delivery (I don't remember if on the first or the second), but they only deliver New Year's Cards. For the most part, people stay home, eat, visit their neighbors, stop in at the local shrine (if you are in Tokyo, there are a couple of shrines which are very crowded at New Years), eat some more, and relax. It's hard to be a type A personality for those three days. I believe that people will go back to work around the 4th or the 5th of January, but the first day they pretty much have an office party.
2. New Year's cards are post cards that everyone sends out to their friends, family and casual acquaintances during December. You can send them whenever you want, but they will not be delivered until after January 1. I always got a few. My Japanese friends literally got hundreds. One important thing to note is that almost all of the post cards are like lottery tickets. They all have a number on them, and certain numbers win prizes. All I ever won was a couple of pretty stamps, but I did keep them.
3. I don't remember a lot about Japanese New Year food, but I do remember that there were two kinds of soups, one sweet and one not. I believe that both had mochi, which I describe as a kind of Japanese rice dumpling (although that is not exactly right, and I'll link to a place where there is a better description.) All the food is the kind that can be prepared ahead, so that everyone works hard on December 31, but no one has to work at all on January 1. The other food I associate with New Year is the mikan. It is not strictly a New Year's food. But mikans are in season in December and January, and I recall spending many a lazy day sitting under the kotatsu, peeling and eating mikans.
4. I also associate New Years with sleeping. It really is a time to relax, and do as little as possible. One year, one of my students invited me to spend the New Year with his family, who lived near Mount Aso. So on December 31, I took the train and spent a few days with his family. All of the children in the family (teenagers) stayed up late on the 31st and went to the temple to hear the tolling of the bells. We all slept until noon the next morning (at least). And, to me, it was like a big slumber party, because, in traditional Japanese style, everyone slept on the floor, on futon, with lots of thick but light kakebuton. (And, I slept in the same room with the children, all of us on tatami mats.)
5. During the New Year, there are non-stop silly game shows on Japanese TV, all the time, along with popular music shows. That's about it. And I learned a couple of Japanese card games, using hanafuda cards,which it seemed as if we played nonstop, while staying in our robes and pajamas. I still have a set of those cards somewhere, although I don't remember the rules of the game any more.
6. Finally, during one New Year celebration a co-worker at the school where I taught invited me to wear kimono. This was a BIG DEAL and an all-day affair. I had to go to the beauty parlor to get my hair done especially (with brillo pad type things in the sides and a lot of teasing and hair spray to get my blonde hair the right shape), and it took some time to put on all of the layers of the expensive silk kimono. (If I get a picture up, please note the long sleeves, which usually are reserved for young, unmarried women.) Then another young worker and I got in her car and drove to the shrine and a couple of other places, to say, "Yoroshiku", which basically means, "Greetings," I think. I'll tell you, it's no mean feat to get into a car wearing a kimono, much less drive.
7. One of the things that happens during the New Year holiday is that people just drop by. When I visited my friends at Mt. Aso, they got into their car one afternoon, and went visiting a number of their friends, bringing little New Years gifts, and wishing blessings.
This is what you say in Japan at New Year's time: Akemashite Omedetoo Gozaimasu!
And so I wish you a Happy New Year to you as well.