I was wondering if Rema seems to pasken Dina D'malchusa dina even for laws between people (not just the laws specific to the king) and there are of many laws that people simply don’t observe. Besides that if we are strict about it we would need to study the laws of the land in the same way we study Shulchan Aruch. Imagine people sitting in shuls and learning the most current laws of the country with a lawyer teaching them: How to throw away the garbage? When and who is responsible about snow cleanup? What are the laws and obligations when you hire a babysitter? There are just so many laws and none of us know them except the lawyers that specify in the specific fields. And we never saw anything like this – serious shiurim regarding laws of the land and making sure you always keep updated. Maybe we could say that most laws are not pedantically observed by the people themselves, so we don’t have a greater obligation than the rest of society? (Also some people say Dina D'malchusa would not apply to democracy but it’s hard to rely just on that isn’t it?)
The source of Dina D'Malchuta Dina is in the Talmud (Nedarim 28a, Gittin 10b, Bava Kama 113b, Bava Batra 54b) Shmuel said, The law of the country is law. The subject was discussed extensively in the Talmud, by the Rishonim and the Poskim. There are many discussions, whether it is biblical or rabbinical, and up to what extent it applies. For example, some Rishonim dispute whether Dina D'Malchuta Dina applies only on laws which benefit the king or does it apply also on regulations and laws that benefit the public. There is another controversy whether this law applies only abroad and with a non-Jewish king or even in the Land of Israel and with a Jewish king. The Rema in Choshen Mishpat 369, 8; mentions a few opinions regarding the application of this rule and he concludes by saying that there are some who opine that Dina D'Malchuta Dina applies in all cases, and therefore one who lends with a pledge can sell it after a year since that is the law of the land, and this is the main opinion. It is however clear that this rule of Dina D'Malchuta Dina applies only on things which the king gains from or on things which are for the benefit of the citizens of the state, but it does not mean that people should judge according to the non-Jewish law system, and it does not apply to anything that contradicts the laws of the Torah (Rema ibid 11, and Shach, Choshen Mishpat 73, 39). Actually, the books of responsa and Jewish law on monetary issues are indeed full of state law in detail wherever it is relevant to Dina D'Malchuta. Regarding your question, why don't we find detailed classes on state law, it seems clear that most important Torah learning is in the Torah itself which is word of the living G-d. It is also very possible that you are right that the laws which are for the benefit of the public that one should follow according to Jewish law, are the famous laws which were accepted by the public and are known to all. In Gemara Bava Batra 55a R. Huna the son of R. Joshua said that even barley in a jar is liable to be seized for the poll tax. The Rishonim (the Ramban, Rashba and Ran) question why did the Amoraim (Talmudic sages) argue regarding the details of the laws of the kingdom, they should have ask the government officials themselves what is the law? They answer, that the rule of "Dina D'Malchuta Dina" is said only regarding laws which are famous and well known ever since as the laws of the kingdom, but temporary regulations and new laws designed to punish the people etc. we do not judge according to them. (Nevertheless there are those who disagree with this. See Magid Mishnah Gezelah 5, 13. Tur Choshem Mishpat 369 In the name of the Rosh. Yam Shel Shlomo Bava Kama 10, 18). According to the Rishonim above, it seems that there is no need to search too much for new laws and regulations and the like, but just to stick to the well-known rules and regulations published in regard to general behavior with accordance the rule of "Dina D'Malchuta Dina". Obviously anything regarding monetary law and ethics of interpersonal conduct one should check with rabbinical judges or rabbis who are knowledgeable in Choshen Mishpat, so they should instruct what is the law according to Jewish law and according to "Dina D'Malchuta Dina" which is customary at that time and that place. Regarding your comment that people say Dina D'Malchuta does not apply to a democracy, this is not true at least according to the opinion of the Rishonim (Rashbam Bava Batra 54b. Responsa Or Zarua 745) that the king's authority regarding Dina D'Malchuta Dina derives from the consent of the people, which exists even more in a democratic regime. And so rule contemporary Halachic authorities (Tzitz Eliezer 16, 49. Igrot Moshe Choshen Mishpat 1, 88. Mishneh Halachot 14, 119. Yechaveh Da'at 5, 64).