I first tried to learn Latin on my own in third grade. I had got my hands on a turn-of-the-century Latin grammar, and was diligently trying to figure out what cases were and why Latin has five or sometimes six, when I stopped trying. Latter, in high school, I got the Latin teacher (who also taught English and German) to teach me Latin during his break period. The deal was I'd get to keep the Latin textbook, as Latin had been cancelled years before to help pay for more footballs or tackling dummies, and I'd give him 5 minutes to smoke a quick cigarette. A couple of years later I took Latin 2 in the local Community College (also taught by a German teacher), and by the time I got to a four year university, I took Latin prose. We read Cicero's Pro Archia and memorized grammatical terms from Buck and Hale's. Other languages I studied formally while in school: Spanish, French, German, Sanskrit, Hittite, Hindi, Homeric Greek, and Tunica. The former Russian teacher at our high school also gave me a copy of the no longer used Russian textbook, and would try to speak Russian with me. He was teaching social studies at that point, but at one point also flew back to Washington DC to translate between Brezhnev and Nixon. After graduating, I also had the pleasure of teaching Latin 1 to four undergraduates at another local four year college as TA for the Latin and Homeric Greek professor (who also taught German).
I think that Latin is a good language to learn because it is different, grammatically, from English, and exposes you to new ways to carve up the morphological and syntactic pie. When some pedant tells you that word order is a feature of universal grammar, you can laugh at them (to yourself). Also, when somebody tries to hypercorrect the plural of virus, you can snort.
Other than Latin, I'd say a good language to study is Mandarin Chinese. It's got no overlap lexically. It's got the tone thing. And it's got a world-class difficult and different writing system. It seemingly has no morphology; at least no inflection. Lots of people also speak it.
