If you want to teach in a public school, you'll definitely need a bachelor's degree, and usually a master's. However, a lot of schools will hire you with only a bachelor's degree as long as you declare your intent to obtain your master's within the first five years of teaching. If you want to teach on the college level, you'll need a Ph.D.
Basically, you need two things. First, you need to be accredited to teach at the level you want. At bare minimum, it means taking a teacher accreditation examination. If you choose, you can major in teaching and complete a practicum (period of student teaching).
Second, you need to be able to speak profusely. Yep, this is one career where it's impossible to talk too much (in anything but English and Pig Latin). Studying abroad is the best way to get the edge on your language learning and experience a different culture first hand. Once you have that experience under your belt and you've gotten so good you've started to dream in Russian (or whatever your chosen language happens to be), you'll need to show your stuff. If at this point you're picturing show 'n tell from sixth grade, let us just assure you that there will be no pet Chihuahuas or slimy socks involved. When the time comes, all you have to do is prove you're awesomely fluent in Cornish pixie, for example. Speak, listen, read, and write your heart out and those language proficiency testers will love you. If you speak more than one language (overachiever, cough, cough), you'll need to prove your fluency in each of them.
One option is to double major in both teaching and a language. You'll be a double stuff Oreo in the job market. And no one can turn down an Oreo.