What college degree should I pursue to learn the fundamentals of the stock market? Are there way to understand how stocks and foreign exchange markets work, including their strategies and drawbacks?
There are several paths of study you could undertake. If you want to learn the fundamentals of the stock market and become a financial analyst, then finance, economics, and accounting (yes, accounting) are all good to study either on your own or in an institution. Furthermore, if you want to study a specific industry, it can't hurt to know a fair amount of the science behind that particular industry. For example, if you want to understand the pharmaceutical or biotechnology industries, knowledge of clinical trials, the FDA's approval process (in the US, at least), off-label uses for drugs, genetic engineering, etc. are all good to know. You don't have to become an expert, but having a firm grasp on the science is extremely useful when evaluating a company's prospects.
If you're interested in becoming an algorithmic trader or a quant, then physics, certain fields of engineering, signals processing, applied math, computer science, or econometrics will get you much farther than a standard finance or accounting degree. Most people can learn the basics of finance; not everyone can learn advanced mathematics.
A lot of the above applies to learning about the forex market as well. Economics is certainly helpful, especially central bank policy, but since the forex market is so massive and liquid, many mathematical tools are necessary because algorithms play a key role as well.
Per littleadv's suggestion, an MBA with a concentration in finance may be an option for someone who already has a degree. Also, an MSF (Master of Science in Finance) or a degree in financial engineering (called an MFE, or ORFE, for Operations Research and Financial Engineering) are other, potentially better options for someone pursuing a more technical career. A high-octane trading firm may not care that you've taken marketing and management classes; they want to hire someone who can understand complex algorithms and design and implement new ones quickly.
Some MSF programs are pre-experience programs, which means that in exchange for taking more time to complete, they don't expect you to have significant work experience in the financial industry. An MBA might require such experience, however.