When doing basic composition in a major key (with no modulations etc.) you tend to just stick to the notes of the major scale, and choose your chords based on that.
With a minor key, it's worth noting that it's very rare to do the same thing. Even music with no modulations or chromaticism thrown in for effect will not be based around 1 "scale" quite in the way that major songs are. Pick up any piece of bach in a minor key, and you'll see this for example. And the same is even more true for modern music, you will rarely if ever find a minor song that sticks solidly to one of the minor scales (whereas sometimes it seems like half of chart music these days is strict major with nothing interesting added).
So, while learning what the scales "natural minor", "melodic minor" and "harmonic minor" are is useful, it's sometimes more helpful just think of minor in general; and know what your note options are within that framework.
To summarise the scales, you have
natural minor (rarely used in classical music, more common in pop music):
C D Eb F G Ab Bb C
harmonic minor (the same but with a sharpened 7th degree)
C D Eb F G Ab B C
melodic minor (both the 6th and 7th sharpened. Notice that this is only 1 note different from a major scale)
C D Eb F G A B C
So what you really have is this (General minor):
C D Eb F G, Ab or A, Bb or B C
So when do you use each note? Generally, the reason that the 7th is sharpened, is because the V-I (perfect cadence) chord progression sound a lot stronger with a major (or dom 7) 5 chord than with a minor one. The 7th note is the third of the V chord, so it changes (in C minor) G minor to G major (or G7).
Compare these 2 progressions. Try playing some arpeggios around these chords:
| Cm | Fm | Gm7 | Cm | (or in A) | Am | Dm | Em7 | Am |
| Cm | Fm | G7 | Cm | (or in A) | Am | Dm | E7 | Am |
The second one probably sounds quite familiar as the "resolution" of a minor song. The first is also nice, but is certainly a more "modern" sound, and is less "functional" in that the chords don't have too much of a strong "pull" to each other. You can imagine hanging out on the Gm for quite a while, and writing some melodic material there.* In the second one though, when you play the G7, you can feel that it wants to go back the the Cm.
Raising the 7th to get this major chord results in a scale that has a big gap in it; (3 semitones!). The top half of the harmonic minor scale can sound quite "middle eastern" or flamenco-y. Personally, I love the sound***, but often we want our melodies to have a "gentler" sound without that big jump, this is basically what the (appropriately named) melodic minor scale is for. It has the sharp 7 to let you have a major V chord, and the sharp 6 to make the melodies sound a little softer**. It rarely affects the harmony, more the melody you play over the chords. You have to be careful though, that if you are currently using a chord with the minor 6th in it (like Fm or Ab major in C) that you don't write a clash between your melody and the chords by using the melodic minor scale at the wrong moment (an A natural over an F minor will sound pretty horrible usually).
Circle of Fifths
So, armed with the information that you have choices with minor in a way you don't with major, how do you decide which "choice" to use when building your chords. In Major, it's easy, you just build the chords starting from the root note and pick the notes that are in the major scale. In the minor, where you get a choice, you can basically pick the one that works best for your purpose. Sure, it's a little complicated, but if you use your ears, you'll land on something that works.
A really common way one is the below*****
| Cm | Fm | Bb | Eb | Ab | G7 | Cm |
That might look a little intimidating, so let's look at it in A minor.
| Am | Dm | G | C | F | E7 | Am |
You can see how it follows the fifths from A until F, and then moves chromatically down until the E7.
Now, why did we choose G and not G#dim (which can be built from the notes available)? Why did we choose E7 and not Em7? Why C and not C augmented? Well, you try it, and see what you think sounds better! When writing in minor, the key thing you have is choices. Now, there are some good choices and bad choices, but those calls are really up to you, the composer! (And probably some theory rules too, but unless you're writing for an ABRSM theory exam, it's really fine to use your ears!!)
Also, notice how when moving up the circle of fifths in a minor key, you quite quickly get to its relative major!. So in Am, when playing around with circle of fifths progressions you will often find you reach C. At that point, all your knowledge of progressions in C major will be useful to you too!
Another very common fifths thing you hear (especially in jazz and latin music) is to use the diminished chord on the ii to lead to the V. So in A minor, you can build a diminished triad (or even better, a four note chord called a half-diminished chord) on the second degree (B D F A in A minor or D F Ab C in C minor). This is usually written ø in jazz (although sometimes it's written as m7b5 which makes it look a lot scarier than it is!)****
| Bø | E7 | Am | or | Dø | G7 | Cm |
This little 3 chord snippet can be used almost anywhere (and jazz composers use it this way). But in a more "traditional context, take a look at this circle or fifths progression. If you left it at C then F, you could believe that you are now in fact in C major. But the Bø E7 Am at the end tells you, no, you're in A minor.
| Am | Dm | G | C | F | Bø | E7 | Am |
You can hear what this sounds like here shortchord.org/BTxRx. Ignore the names of the chords on that website, they are wrong. Some of the chords here have added notes, but then don't need them; I just wanted to show some possibilities for the future. Also I love 6 chords at the moment; so playful sounding!
| Cm | Fm7 | Bb6 | Eb | Ab6 | Dø | G7 | Cm |
which is really the below but with a bit of flavour:
| Cm | Fm | Bb | Eb | Ab | Dø (or just Ddim) | G7 (or just G) | Cm |
Pure circle of fifths.
*In fact, some modern music does this very thing. Street Spirit by Radiohead and ain't no sunshine when she's gone, both use the minor 5 in that way; it doesn't pull anywhere particularly strongly, it just hangs out where it is. This device isn't in classical works much at all.
**For example, look at "ne me quitte pas" also "if you go away" in English. The ascending line melodic line just before the chorus comes in ("but if you stay" in English, but the Jacque brel version is better haha) pretty much outlines how the melodic minor notes can be used to great effect. These notes lead up to the tonic minor chord; the melody would sound very different if the harmonic minor were used.
***A more extreme version of this has the big jump at both ends of the scale. C Db E F G Ab B C . Used a lot in Arabic music, and also in that song from pulp fiction. Huh, huh haaaaaaaaaa.
****You probably know that jazz progressions tend to go all over the place and end up in strange keys. This little progression has such a strong pull, that it can be used to to pull back to the minor key, and tell the listener where "home" is again, even after you've been lost somewhere else. In fact, bossa nova composers love it so much that they sprinkle it throughout their songs to lead into minor chords; even outside of its "proper" context.
***** Off the top of my head, muse's "unintended" basically uses this progression in E minor, except the E minor is substituted for E major (but the other chords are taken from the minor. It's pop music; you can break the rules if you want).