What arrears of laws would be best to investigate if I am trying to open a original design business (flyers, graphic tees, paper products)?
Contract law,Business law, Copyright law?
Specific examples of classes online or at community colleges?
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There are basically three kinds of college classes about law that are available for undergraduates: constitutional law (normally in a political science department), business law (normally in a business or management department), and criminal justice law (normally in a criminal justice associate's degree program aimed at people seeking law enforcement careers).
Less often there are classes on copyright law in journalism programs, or a class I took in law school called "practical law" that basically focused on consumer protection laws. Sometimes real estate law will also be taught as part of a certificate program for prospective real estate agents.
For someone seeking to start a small business, a business law class would be most beneficial of those available to undergraduates. The constitutional law and criminal justice classes would provide little of value to a small business owner. The copyright law class would cover lots of irrelevant issues, not many relevant issues and is fairly easy to figure out without formal instruction compare to other topics that would be studied in a business law class.
Few schools teach "practical law" or real estate law, and as a prospective business owner, a business law class would be more useful than either a "practical law" class or a real estate law class.
A business law class will be most helpful to you because it will get you to start thinking about all of the aspects of a design business that are more about business and less about design, which people who start businesses in creative fields are prone to overlooking.
Also, even for prospective lawyers, the main skill that you learn in a formal educational setting is "issue spotting" which helps you figure out when you are dealing with a legal issue and need to think about it as such, rather than to provide definitive answers, because the law changes frequently anyway. A business law class is best suited to help you to develop "issue spotting skills."
Furthermore, any really serious copyright law issues is something that a single college course would not qualify you to handle appropriately in any case without the assistance of a lawyer.
You do have a tag for "tax law" which is often taught in an accounting department of a business school. But, tax law changes very rapidly, and isn't very useful unless you know a lot of it. Tax law also is quite easily outsourced to CPAs and tax preparers.