Pope Gregory XIII established the new calendar 24 February 1582. The Gregorian calendar replaced the Julian calendar, except, of course, in communities that refused to accept it. Julius Caesar, as part of his calendar reform, established January first as New Year's Day; Pope Gregory accepted this convention.
But in various times and places people marked the new year on other days -- for example, in England before the Norman conquest, the new year began on 25 December or 25 March, from 1087 to 1155 on 1 January, and from 1155 to 1752 on 25 March, though people still spoke of January first as New Year's Day, and after 1600 in Scotland the new year officially began on January first again. (England did not accept the reform until 2 September 1752; because of the reform, the next day in England and its colonies was 14 September 1752.)
This calendar abides by the Gregorian convention (and the Julian convention as well) in that it always starts the new year on January first. If you use it to date manuscripts from communities that marked the new year on some other day, beware.
A wikipedia article describes the Roman Calendar. For more information on Roman numerals and dates, see Paul Lewis's page.
To find the date of Easter in any given year, try the calculator at the U.S. Naval Observatory or the help provided by the Astronomical Society of South Australia.