A very rough breakdown of the Japanese school/work year with holidays are:

January 1: New Year's Day 
January 15: Seijin no Hi
You come back from Winter Vacation around the first (work) or second (school) week of January. 

February 3: Setsubun (not National Holiday)
February 11: Kenkoku Kinenbi 
February 14: St. Valentine's Day (not National Holiday)
This is usually around the time students start to cram as this season is also known as "exam season." High school and University entrance exams begins around this time. So does job interviews.

March 3: Hinamatsuri ("Girl's Day")
March 14: White Day (most definitely a holiday made by corporate-commericialism)
March 20: Vernal Equinox
Cherry blossoms start to bloom. Graduation season around the end of March. Varies by school, but usually second or third week in March. You have Spring Break until the next school year starts, which is...

April 1: Start of Japanese Fiscal Year. So if you were in the second year in high school at the end of March, you come back from Spring Break and you are a third year student. This is usually also the time where you are officially become a corporate worker if you were selected for hiring. If you already work, this is the time you get a promotion/pay raise. 
April 29: Midori no Hi
Green Day marks the first day of several consecutive National Holidays to early May. Rather than having you coming to school or work one day and not the next, they call this "Golden Week" where the schools and work give the whole week off.

May 3: Constitution Day
May 4: Kokumin no Kyujitsu (since there was a National Holiday on the 3rd and the 5th, the government decided to create this holiday so people can have the entire week off)
May 5: Children's Day (Boy's Day)

June - no holiday, but a very humid summer season begins

July 7 - Tanabata (not a National Holiday)
July 20 - Umi no Hi (Ocean Day)
July is also known in Japan as Typhoon Season. This is when several typhoons start to strike Japan. Students begin summer vacation usually around the third week of July. As most people would know, this is the time summer festivals (fireworks, etc) begin - starting with Tanabata. 

August - no holidays. However, along with July, this month is filled with summer matsuris and Obon festivals. School usually resumes end of August, early September

September 15 - Keirou no Hi (Respect-the-Elderly Day)
September 23 - Autumnal Equinox 

October 10 - Taiiku no Hi (Sport's Day)

November 3 - Bunka no Hi (Culture Day)
November 15 - Shichi go san (not a National Holiday)
November 23 - Kinrou Kansha no Hi (Labor Day)

December 23 - Emperor's Birthday
December 24 - Christmas Eve (not a National Holiday)
December 25 - Christmas (not a National Holiday)
December 31 - New Year's Eve
Winter vacation usually begins before Emperor's birthday, usually on the third week of December.