Look for local classes in Korean and Chinese at community colleges or universities near you. Since learning a language by yourself is very difficult, being taught by a fluent speaker is the best way to start. If you are in the Washington, DC area, Northern Virginia Community College teaches Chinese in most academic sessions.
Try your local public schools if you cannot afford to take college courses; some public schools offer Chinese and Korean classes in their adult and continuing education programs.
Use Chinese and Korean language books if you cannot afford to take a class or do not have access to language classes. Local bookstores will have language-learning books, as will public libraries.
Use your Internet access to connect with online Chinese and Korean language services; websites such as Mando Mandarin offer Chinese language services and online Chinese tutors, while Verbal Planet offers the same but in Korean.
Apply to the Middlebury Chinese School to spend 9 weeks in complete Chinese immersion. Even though the program lasts only for one summer, you have to sign a language pledge stating that you will minimize your use of any language other than Chinese. In the process, you will progress, on average, an entire year in just 9 weeks.
Travel to China or South Korea. There is no better way to learn a language than to immerse yourself in it. Note that U.S. citizens can travel to North Korea, but only during the Arirang Festival in September. Be aware, however, that you will not be allowed to interact with many North Koreans.