The day is the third of the Four Pillars of Destiny and in Chinese fortune telling represents information about the person him/herself, his/her adult and married life.
The sexagenry cycle was used in China since the second millennium BC (Shang Dynasty), as a means of naming days (just as western cultures use the days in the week). This use of the cycle for days is attested throughout the Zhou dynasty. More recently this is not as popular but is still used in Almanacs and calendars.
The 1st day of a new year in the sexagenary cycle should be the Lichun (節氣 lìchūn). The Lichun is the 1st solar term. It begins when the Sun reaches the celestial longitude of 315° and ends when it reaches the longitude of 330°. It more often refers in particular to the day when the Sun is exactly at the celestial longitude of 315°.
In the lunisolar calendar, New Year's Day might be before or after Lichun. A year without Lichun is called 無春年 wú chūn nián (no spring year). 無春年 is also known as 寡婦年 guǎfu nián (widow year) in northern China or 盲年 máng nián (blind year) in southern China. Marriage is believed to be unlucky in a year without Lichun.
I have found it hard to find the corresponding elements and animals of the days themselves, however you can use the calculator to find out your day on my post about the Four Pillars...