Many things happen in Italy in the month of March.
First of all, because Italy is in the northern hemisphere, Spring start in the month of March and temperatures start to get milder. It is not the absolute best time to visit, but it is a good time for the central and especially the southern regions.
Father’s Day is celebrated during the month of March, on the 19th. The tradition calls for special sweets to be prepared on that day. They have variations in many regions, but generally they are beignet filled with an egg based sweet cream. They are called “Bigné di San Giuseppe” (St. Joseph's Day beignets). They are called that way because Father’s Day happens the day when St. Joseph is remembered.
Finally, on March 8th Italians celebrate International Women's Day. People give mimosa flowers to women and girls as a symbol of respect and appreciation.
You can learn the Italian calendar with months and seasons here.