Learning the piano is the best way to learn about music in general. Its orderly, logical keyboard layout reinforces music theory concepts that can be otherwise difficult to learn and appreciate. Plus, the ability to play bass, melody, and chords simultaneously make it a great way to learn how all the different parts of music work together to make a whole sound. In fact, it's such a good way to learn about music that most music schools require their students to pass a keyboard proficiency test, regardless of the instrument they're primarily studying. It's not an accident that almost all of the major composers of Western classical music were at least decent keyboardists, and some (Beethoven*, for example) were among the very best pianists in the world.
If you were also a bassist, I'd recommend too that you learn to play the drums at least passably well. The bass/drums interplay is the foundation of most pop music, and when both instrumentalists understand each other better, it can go a long way to making your lives easier.
As for trumpet, saxophone, and other non C-based instruments, you don't have to learn how to play them, but it would probably be helpful to at least understand that they call the notes by different names than you do: for example, when you ask a trumpet player to play a C, the sound that comes out will actually be a B♭. Understanding that will clear up a lot of confusion.
*Possibly apocryphal but believable story about Beethoven's piano prowess: He played the piano for the world premiere of his Piano Concerto #1 in C Major (this part is undisputed fact). When he arrived at the concert, the orchestra had tuned to the oboe, as is SOP, so they were in standard concert pitch. But the piano, it turns out, was a half-step flat. Rather than re-tune the orchestra---because, after all, he wrote the piece in C and not B for a reason---he proceeded simply to sit down and play the entire piece in C♯ (!!).
Update: The Beethoven story apparently comes from Carl Czerny, who was a student of Beethoven's at the time and who had been asked by Beethoven to serve as page turner for the performance. To my mind, this lends credence to the story, as astounding as it is.