No. F minor is the starting point. So your key signature would look like this...
Four flats which is also the key signature for the relative major Ab. A minor key is actually a bad example for my point (I'll mention why) so I'll use Ab major. In a song written in Ab, you'll see the Ab major chord along with related chords that are made of notes in the Ab major scale ( I ii iii IV V vi vii ). Now if you're listening to pop or something, you probably won't hear much outside of those several chords
. In actual music though, the composers throw in other chords too. If I really wanted to I could throw a D minor chord in the middle of that song in Ab. It wouldn't necessarily sound good, but I could do it. With that said, in a lot of classical music you'll see chord progressions that include chords from other keys even though there is no explicit key change. These sound good and TBH those chord progressions are what make classical and similar music so beautiful. It just burns me that popular music doesn't use such interesting progressions.
A simple example of this. I could start out on Ab major chord, then play the IV chord with is Db major. From there I could decide instead of playing, say the V chord, I could play the IV chord in the key of Db major which is Gb major. Then I could play the ii chord in the key of Db major etc. and then eventually return to chords in the key of Ab major. This doesn't have to be a key change. Sometimes it's only a chord or two taken from another key. Note that you'll see a bunch of accidentals on the sheet music when this happens. Sadly, when you hear a key change in pop, it's usually just transposing the same exact thing a couple steps up *facepalm*. (btw, it is easy to hear a key change like this.)
The composer can also decide to throw in whatever notes he wants. For example he could use whatever chromaticism in the melody he wants even if those notes don't fit into the chords being played. In the words of my piano teacher, there are no wrong notes.
I decided not to use a minor key as an example because there's something funny that goes on in minor keys. The natural minor uses the same notes as indicated in the key signature, but the 'harmonic minor' scale has the 7th (note in the scale) raised one half step (the harmonic minor is commonly used). So in a song in F minor you'll probably see a bunch of accidentals and many of them will be E natural. To confuse things even more there's also a melodic minor scale which is slightly different. In the song, sometimes you'll see the E natural, other times you won't. You may even see a D natural which comes from the melodic minor scale. All of those are perfectly acceptable and still in the key of F minor.
Knowing all this makes looking at a piece of music much less daunting. Instead of looking at a chord with a bunch of accidentals and needing to go through and look at each note, you'll start to recognize the pattern. This is actually one of the most helpful things in reading music even when there aren't accidentals.
Now that you've had a crash course on music theory, find a book or an appendix in one of your books that has all the scales, chords, and arpeggios. This is known as the circle of fifths/fourths. Practice them and this will make much more sense
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You can close your eyes and you're on a beach, with the band playing right in front of you.