For people with "hands-on" experience in mathematical research, what languages are the most beneficial to learn? I know that many graduate programs require some degree of non-native lingual proficiency, so what would be the pros and cons of pursuing various languages?
(I realize that this question is a bit broad; please ask if anything needs to be specified.)
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French for sure. Opens up a whole wide world. Much more opportunities. Non-francophone countries accept to do business in English.
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Not sure it is an answer... let's try!
If you search to learn a new language just to enhance your efficiency to do research, I think you're loosing your time. Even in countries that had a strong tradition of writing in their own language (France, Russia), most publications are in English nowadays. Important books in Russian are quite quickly translated into English. The material contained in old articles can be found in more recent work (in my specific topic, everybody quotes a certain article in German by Hopf, by I guess few people read it). Except for history or philosophy of mathematics, the mathematical benefit of reading the original paper is very low. But learning a new language is time-consuming! If you want to be an efficient article-reader, use that time to read maths in a language you already know.
The only reason you should learn a new language is the same as non-mathematicians: learn it because you are genuinely interested in it! Of course, one interest could be being able to read the work of your favorite mathematician in its original form, but don't expect a huge impact on your career. At least, learning a new language may ease your collaboration with foreign coworkers, but it is kind of an indirect impact.
Most US-based PhD programs have, as a requirement for the degree, at least having reading proficiency in one of: French, German, Latin, Russian or, increasingly, Chinese. The first few for obvious historical reasons. In fact, except for Chinese, the impetus is predominantly historical nowadays; internationally, most cutting-edge mathematics is being published in English.