I mentioned before that I didn't really think my time playing the saxophone in 6th-9th was worth it. I've thought further, and have come to some conclusions about what set of instruments a well-rounded musician should study at minimum as they come up through school.
If you want to understand western music, you've got to understand harmony, which is a much easier thing to do when you play a polyphonic instrument (i.e., one that can play multiple notes simultaneously). Nothing beats a keyboard for clearly laying out the relationships of pitches to one another, or the black/white key configuration for illustrating the main scales in our sonic vocabulary. Other traits that recommend the piano are a good dynamic range, plus a very transferable skill set; if you can play a piano keyboard, you can pretty much a play any synthesizer or organ, as well as more obscure instruments like harpsichords and harmoniums.
The guitar is another solid contender to fill the harmonic role, but I think that the variety of tunings and the fretboard structure provide a more convoluted model of the relationships between pitches and scales. Like the piano, the ability to play the guitar is transferable to a myriad of stringed instruments, from the bouzouki to the oud, not to mention the rock mainstays of electric guitar and bass.
Plucked and hammered strings can be very moving, but there is no more expressive instrument than the human voice. Even if you have no aspirations of singing long-term, taking formal voice lessons will likely teach you more about subtle expressiveness in a melodic line than any instrument. To get the most out of studying voice, though, you must learn to sight-sing. With an instrument, it's possible (and in my experience, common) to learn to read music by converting written notes directly into physical actions (keys pressed, breath exhaled), without really learning what specific intervals and chords really sound like. That is, you can sort of inadvertently bypass the hearing part of the process. Every college music student is expected to be at least minimaly competent at transcribing and playing back what they hear. At my school, there was a class called "aural learning"; practically every singer in the class breezed through with an A, while many instrumentalists struggled to pass.
Finally, any decent musician is going to be expected to perform as a member of a group, whether it's a drum line, symphony orchestra, or power trio. There are any number of ways you could go here, but I really think violin is the best choice (and, by extension, a school or community orchestra) - it's useful in orchestras, country music, and the occasional rock group. There also seem to be more paying gigs out there for violin than any instrument save piano (and most of those are accompaniment jobs).