i.e do we refer to the key of F# more than the key of Gb or are they truly interchangeable and referred to equally? Or does the context matter?
(I’m trying to get a clear hold on what keys are more meaningful to know by heart in terms of signature, scales, etc).
As Tim notes, the usual approach is to use the key signature that leads to the simplest accidentals. Thus, you'll see D♭ major rather than C♯ major and B major rather than C♭ major. A prominent example of this is Beethoven's Moonlight Sonata, which is in C♯ minor. The middle movement is in the parallel major key, but to simplify the notation, it is in D♭ major.
F♯ major and G♭ major both have six signs in the key signature, so you can go either way, but with their relative minor keys you do see E♭ minor more frequently than D♯ minor because of the tendency to raise the sixth and seventh degrees of the natural minor scale: it's simpler to read naturals in E♭ minor than double sharps in D♯ minor.
There are exceptions, of course. Bach's Well-Tempered Clavier is an interesting one: in addition to the fact that he uses C♯ major instead of D♭ major, the eighth prelude of book 1 is in E♭ minor, but its fugue is in D♯ minor (perhaps because it is a transposition of a fugue in D minor). In book 2, the eighth prelude and fugue are both in D♯ minor.
Another place where you'll find enharmonic equivalents is in parts for transposing instruments. For example, an E♭ saxophone part for a piece in concert E major will typically be written in D♭ rather than in C♯.
May be dependant upon the instrument played. Some music is easier to read one way, some the other. But basically, simplicity wins most times. When it comes to keys C♯/D♭ for example, C♯ has 7 sharps against D♭ with 5 flats. So it could be argued that 5 is simpler than 7 to remember while reading.
With G♭/F♯, though, it's often six of one, half a dozen of the other! So brass would most likely prefer flats, while strings, maybe, sharps.
There are occasions where certain notes, though (not key signatures, note) will need to be specified. Take, in key C, G♯/A♭. It will depend on what harmony or function that note has. For example in chord C+, it has to be G♯ whereas in Fm it'll be A♭. In other circumstances, it's usual to use sharp accidentals in sharp keys, flat accidentals in flat keys. Sort of easier to keep one's mindset in one or the other. Although, frequently, I come across pieces where the chord should be spelled one way, but the dots show the other. A little confusing!
EDIT: In answer to you and which keys you should get better acquainted with - it is rather instrument orientated. With piano - all, with others, the obvious starter will be C - even if you play a transposing instrument. Then move through the circle of 5ths, both ways.
AND - by the time you are conversant with 5 or 6 of each in the key signature, you won't be concerned about another.
The short answer to your title question is: They do at times. For one example why, let’s look at F#/Gb (6#,6b) and its circle of 5ths neighbors C#/Db (7#,5b) and B/Cb (5#,7b).
Often in traditional classical music there are modulations. One of the most common is to modulate to the key of the dominant, or the 5th degree. If you are in B then it makes more sense to modulate to F# than Gb because enharmonically F# is the 5th of B. Gb would actually technically be the diminished 6th of B, not the 5th. For the same reason, if you are in Gb it makes more sense to modulate to Db than C#.
Another reason enharmonic key spellings may matter is when using transposed instruments. If you write a piece in E and have a part for say, a Bb trumpet, which sounds a step below where it is written it makes sense to put it in F#, a step above. This is not always done for practical reasons, for example transposing from C# to D# (9 sharps!) but in this instance it would make more sense to have your keys be Db transposing to Eb.
My advice is to learn the keys from zero sharps or flats and gradually work up to 7 sharps(C#) and 7 flats(Cb). Those two keys are all the notes of C major either sharped of flatted. You will almost never see anything beyond seven. The process of learning them is additive and both the sharps and flats are added in 5ths, up for sharps (F,C,G etc,) and down for flats (B,E,A). If you study the circle of 5ths you will start to see that there is a logic to it and everything has a pattern.
I'm actually not sure... I think that strictly speaking enharmonic keys aren't truly interchangeable under all circumstances (not when one is dealing with instruments that can make the slight difference in pitch sound out loud), but the key is a choice on the composer's part, and there's no benefit in using a key with a complicated signature over a simple one.
In practice, keys with either double sharps or double flats are extremely rare, so there's little reason to learn them in case you might run into one many years from now when you won't remember it anyway. (Not to mention keys that verge into the realm of triple sharps/flats or higher - these really exist only as theoretical construct.)
You will meet Eb major. You're very unlikely to meet D# major. You won't meet Fbb major in your life.
I would say there are two general points to guide you:
Use the enharmonic spelling that will provide the simplest (least number of sharps or flats used in the) key signature. So, for example, use D flat major with 5 flats versus C sharp major with 7 sharps.
Use the tonic pitch letter which makes harmonic/intervallic sense. For example, if you were in B major - five sharps - and then modulate to the dominant - which will be based on a tonic one perfect fifth above B - the then new key tonic will be F sharp, not G flat, so write in F sharp major.
Notice that in a case like Bach's Well Tempered Clavier, the pieces in C sharp major could be in the simpler key signature of D flat major, but the sequence of keys for the pieces is by chromatic ascending half steps, which will normally use sharps for an ascent or flats for a descent. I suppose that type of choice could be a third category.
