You're partially right about the transposing instruments, but I think you might be a bit confused about how this affects the keys they are playing in.
A trumpet is a Bb transposing instrument - write a C on a piece of trumpet music, and the note they will play will be Bb concert (a Bb on say, a piano). This means if you want the trumpet to play C concert, you have to write it a tone higher - D on the trumpet notated music.
So, for a piece of music in C concert, the bass, guitar, piano and trombones (and some other instruments) are written in C, trumpets and tenor saxes are playing in D, alto and baritone saxes (Eb transposing instruments) are playing in A.
For a piece in D concert, alto and baritone saxes would be playing in B - 5 sharps! A bit tricky for them. So this is why arragements for this sort of ensemble are often in "flat" keys (concert). For example, a piece in Bb concert would have trumpets and tenor saxes playing in C and alto and baritone saxes playing in G.
By the way, guitar and bass are also transposing instruments, but the notes they make are transposed by a full octave (compared to their written music) so this doesn't mean they need to be written in a different key signature to the piano.