I'm trying to play the chords for a songs chord progression.
It goes: Gb, Bm/Gb, Gb, Bm/Gb, Gb, Bm6
First, it’s important that you understand that there is absolutely no rule that says that a song has to use only the 7 diatonic notes of the key it is written in. Any note may be used in a chord or a melody at any time during a song.
As for your specific chord progression of Gb to Bm/Gb, the main problem with this progression is a poor enharmomic spelling. What that basically means is it is using accidentals in an illogical way, in this case, combining flats and sharps when there is no need to do it. Also, Gb (or F#) is an awkward key to begin with.
As it is, it’s confusing because you have a Bm, B-D-F# with a Gb in the bass, which is also F#, the 5th of the Bm chord.
It is easier to think of this as F#, in which case the second chord would be simply Bm/F#, or a IV minor chord over I in the bass.
However if it has to be in Gb then the thing to do is think of it as Cbm/Gb. In this scenario, Cbm is Cb, Ebb, Gb and you’ll notice the only note that is not in the key of Gb is the Ebb (the same as D).
What are the notes I play on a piano for Bm/Gb? Does is mean play both Bm AND Gb - or - Bm OR Gb?
This is "slash chord" notation. It means play the chord to the left of the slash over the bass tone to the right of the slash. Play the chord Bm over the bass Gb.
Bm = B, D, Gb
This system of chords uses what's called tertian harmony, chord built of stacked thirds above a root. You should learn to recite the gamut of letters in thirds - B D F A C E G B D... - and then you apply accidentals to set the appropriate intervals major thirds, perfect fifths, etc.
Bm is B D #F.
The alternate options of Gb or F# are called enharmonic equivalents. They are the same pitch, but enharmonically spelled differently. Chords should be spelled by what's intended. Bm is B D F# but B D Gb would be an odd augmented triad with an augmented third, surely not the intended chord.
...not in the scale of Gb major...do I need to play the closest "legal" notes that fit the scale?
No. Play what is written.
But in this case it will be a lot easier to talk about if the passage if first re-written to spell the chords correctly...
F#, Bm/F#, F#, Bm/F#, F#, Bm6
...and the scale will be F# major.
If you were to talk about this being in the key of F# major, you could then say the diatonic chord in F# major, rooted on B is a B major chord. However, in F# minor, the diatonic chord rooted on B is B minor. F# major and F# minor have the same tonic - F# - and so are called parallel keys. And, you can "borrow" chords between parallel keys. You can "borrow" Bm from F# minor and play it in F# major.
If you identify the chords with Roman numerals, it would be F#: I iv6/4 I iv6/4 I iiø4/2. The chord root B is the fourth degree of the scale so it's labelled iv. So, the progression is in F# major and borrowing the iv chord from the parallel minor.
Chord symbols like Gb, Bm, or Bm6 are literal, not relative to the key of the piece as a whole. So no matter what key a piece is in, Gb always comprises the pitches Gb Bb Db, Bm comprises B D F#, and Bm6 comprises B D F# G#.
The slash notation, in this case, indicates that Gb should be played in the bass. That is, Gb should be the lowest pitch played. It need not be part of the chord, though in this case it is (Gb = F#). Thus, Bm/Gb means "play a Bm chord and also play Gb in the bass."
For more on slash notation, see What does the chord notation X/Y (“slash chord”) mean?.
What are the notes I play on a piano for Bm/Gb?
The notes you play for Bm/G♭ are the same on any instrument. The symbol to the left of the slash indicates a chord. In this case, it's a Bm chord, which is B, D, and F♯. The symbol to the right of the slash is the bass note, which may or may not be a note of the chord on the left of the slash. In this case, that note is G♭, which is enharmonic to F♯. That means G♭ and F♯ share the same pitch (in equal temperament) but are named differently. You're correct that B and D are not in the G♭ major scale. Neither is F♯, for that matter.
Do I need to play the closest "legal" notes that fit the scale?
You play the chord indicated. It is common for non-diatonic chords to occur in a piece of music. That means that a piece of music in a particular key can "borrow" chords from another key. It doesn't necessarily--or even usually--mean that the key changes.
However, it is likely that whoever notated this music is confused. In the key of G♭ major, you would usually call this chord C♭m. A C♭ minor chord is C♭, E♭♭ (E double flat), and G♭. C♭ is enharmonic to B, E♭♭ is enharmonic to D, and G♭ is enharmonic to F♯. In this case, there would only be one note not in the key... E♭♭. You can extend the meaning of enharmonic to chords, meaning that their constituent pitches are the same as a differently-named chord. Therefore, C♭m/G♭ is enharmonic to Bm/F♯ and Bm/G♭.
