What weak and beggarly elements had the previously non-Christian Galatians been observing?
Good question. The context gives a general, but not a specific, answer.
Galatians 4:3-10 KJV — Even so we (I think this means "the Jews"), when we were children, were in bondage under the elements of the world: But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law, To redeem them that were under the law, that we might receive the adoption of sons. And because ye (and this is directed at the Galatians, and Gentiles in general) are sons, God hath sent forth the Spirit of his Son into your hearts, crying, Abba, Father. Wherefore thou (individual Gentile) art no more a servant, but a son; and if a son, then an heir of God through Christ. Howbeit then, when ye knew not God (before the Gals knew Christ), ye
did service unto them which by nature are no gods. But now, after that ye have known God, or rather are known of God, how turn ye again to the weak and beggarly elements, whereunto ye desire again to be in bondage?
Ye observe days, and months, and times, and years.
The latter bolder text is what they were attempting to do as Christians, only following Judaizers, but it suggests that they used to do something similar in their pagan idolatry, to false gods. I don't know a whole lot about the Greek or Romance religious observances, but I suppose they had holy days (holidays) where they were supposed to offer sacrifices or do something different from normal days.