In Japan, either a man or woman confesses his/her love and asks "つきあってください (Will you be my girlfriend/ boyfriend?)" to be a couple in many cases. 

Different from European or American culture in which people date several times and get to know each other before becoming a couple with an unspoken agreement, in Japan it is not until you confess your feelings and get an 'OK' that you start seeing each other. It may be Japanese to want to clear the date or timing when you start seeing each other. That's why the confession, the day you confess your love, is a big event for Japanese. 

"つきあってください" literally means "Will you go out with me?".
It is even not a rare case to ask, "つきあってください" before dating and date for the first time after becoming a couple.
Some people don't like to hold hands or kiss unless they are a couple, so you should understand the culture which is important for you to tell your feelings before doing such things in Japan.

Since Japanese couples know when they start seeing clearly, it is common for them to celebrate the day they started seeing as an anniversary every year. Some couples celebrate it every month like a 1-month anniversary or a 2-month anniversary. I think that is too much, but it is common to celebrate in multiples of 3, like after 3 months and 6 months. 

Generally speaking, women place more importance on their anniversary than men, so it is common for a boyfriend to forget their anniversary and his girlfriend to get mad. Some girls set trivial anniversaries like "first hold hands anniversary" and "first kiss anniversary". When girlfriends say, "What day it is today?", it is a very frightening question for men. 

On an anniversary they go out for dinner and give presents to each other. How much you spend on it depends on one's sense of value of the anniversary. It is better for you to discuss it with your boyfriend/girlfriend in advance.