There are two kinds of numbers you can use to talk about a specific day: an ordinal number and a cardinal number. Cardinal numbers represent amounts like one, two, and three. Ordinal numbers represent a place in a series like first, second, and third. I think of cardinal numbers as the numbers you see on playing cards.

When you’re writing out a date like January 1, 2023 (in the American style), the day is written as a cardinal number. So you should never write January 1st, 2023. The weird thing though is when you’re speaking, even though it is written as January 1, you say, “January first.” So when you are reading a date that is written January 1, 2023, you say “January first, two thousand twenty-three.” That’s probably why a lot of people get confused about how to write it.

The instance in which it is OK to use an ordinal number is when you are writing the 1st of January, because you are placing the day in a series: of all the days in January, this day is the first. For example, your invitations could say, “Please join us for a party on the first of January.” In that case, it’s correct to use the ordinal number, first.
For more grammar tips for writing out dates, listen to this episode, or read a transcript here.