While most schools don't start foreign language training until junior high or high school--a time when it's too late to develop fluency--some schools offer foreign languages from elementary school. You typically don't have much choice in what language your child will learn--it's usually Spanish or French--but this can be an excellent resource. Look to see if schools in your area offer language training. This is more common in private and alternative schools.
If you can't find language classes during school hours, you can take your child to a language learning specialist. Some companies, such as The Language Workshop in New York City or Little Ambassadors in Hawaii, offer classes in a variety of languages. Once-a-week training may not be enough to gain fluency, so schedule multiple classes a week or extra study time at home.
DVDs are a great way to introduce kids to a new language. Shows such as Dora the Explorer or Ni Hao Kai Lan are primarily in English, but introduce a few words in foreign languages. You can also find DVDs that are specifically designed to teach a foreign language to children, such as the Little Pim DVDs or the BBC's MUZZY.
Software programs are another good way to teach a foreign language to your child. The Rosetta Stone software is a good choice for kids aged five and up. Jump Start also makes language-learning computer games for kids. Depending on the language that your child is studying, you may also be able to find free games online.