How should we observe Day of the Atonements / Yom HaKippuriym / Yom Kippur?

Answer From The Annual Reading Schedule - Week 28 Notes For Leviticus: First of all, for any observance we must remember that Yahweh does not like iniquity or unrighteousness and assembly to be mixed together (Isaiah 1:13). What this means is that we cannot be hypocritical. We can't be pretending to follow Yahweh when it comes time for an observance, and then on all other days living wickedly without any regard for the laws of Yahweh. We can't oppress the poor, the fatherless, and the widow a day before a feast, keep the feast and rest, and then go back to oppression the day after the feast and expect that Yahweh will be pleased with us.

We need to make sure that we are not pretending and that we are looking at the way we live and our hearts to see if they are in line with Yahweh's laws. If we're just pretending on feast days and other observances and not really following Yahweh in the rest of our life, then He will not be pleased with our fake attempts to keep His Set-Apart Days. Now, to the specifics of this question:

The Day of The Atonements (Yom HaKippuriym commonly called Yom Kippur) is mentioned in this reading in Leviticus 23:26-32. The Day of The Atonements is a day to remember how Yahushua made atonement for us with His own blood. In Hebrew the word Atonement is related to "covering over" – specifically covering over our sins. As so many of the set-apart days do, Yom HaKippuriym also points towards Yahushua and reminds us of what Yahweh has done for us through Him.

We are to rest, not work, have a Set-Apart Gathering with a Reading of the Scriptures, focus on prayer, confession of sin, and repentance, and afflict our being. Afflicting our being means to refrain from pleasurable things such as marital relations, food, and water. So it is a commanded day of fasting and afflicting ourselves. Also, please see the answer to the question, "What are we to observe on the 10th day of the 7th Biblical month and how do we observe it?" in the Reading notes for Week 27 of the Annual Reading Schedule. Those reading notes also help answer this question as well.