This online calculator finds your full (normal) retirement age based on your birth year, then calculates your retirement date by adding retirement age to your birthday. Retirement age is defined (as of Feb 2020) based on the Full Retirement Age page on Social Security Administration site.
In short, people born before 1938 have the age of 65 years, then retirement age increases by 2 months each year up to 1943. People born between 1943 and 1954 have the age of 66 years, then again it increases by 2 months per year up to 1960 to finally become 67 years.
If you were born on the 1st of January, the year to determine your retirement age is the previous year.
You are eligible to receive full social security benefits the next month after your retirement date unless it's the first day of the month. In this case, you are eligible for the same month. Since the Social Security Administration pays individuals a month behind, you will receive the first benefit payment yet another month later after your retirement date.
Finally, your application for benefits could take up to three months to be processed, so you'd better apply 3 months beforehand.