Galatians 3 (and the last verses of chapter 2, and the first verses of chapter 4) tells us that we are free from the "law". But I wonder what law are we speaking about here? The old Jewish laws, the 10 Commandments, or something else?
This refers to the Law of Moses which would include the 10 Commandments. In Galatians, Paul is refuting the Judeizers who believed in Christ as Savior but also believed that the Christian was obligated to observe all the Law of Moses. See Acts 15; The Jerusalem Council was formed to answer the question, should the Gentile Christians be circumcised as the Law of Moses required? Acts 15:7 “But there rose up certain of the sect of the Pharisees which believed saying that it was needful to circumcise them and to command them to keep the law of Moses.”
This is mostly referring to "the Law" as the Torah, the five books of Moses.
But it is a bit complicated:
The word "Torah" in Hebrew is derived from the root ירה, which in the hif'il conjugation means "to guide/teach" (cf. Lev 10:11). The meaning of the word is therefore "teaching", "doctrine", or "instruction"; the commonly accepted "law" gives a wrong impression. Other translational contexts in the English language include custom, theory, guidance, or system.
This is not entirely correct imho, since it very well also means the mosaic laws, for which the 10 Commandments are just one example. So, "law" just narrows it down to much. This is a traditionalised translation mixup from Hebrew over Greek to English.
Paul just using the word "law" is routed in the tradition started at least with the septuagint. Paul wrote in Greek and the septuagint renders Torah as well as profane law just as the Greek nόμος (nomos).