I am currently writing a piece of music for fun. In it I use a I-IV-V (C-F-G) chord progression but then I looked up other good chord progressions that began with C and found (C-G-Am-F) and was wondering if I could combine them to make (C-F-G-Am-F).
If you're new to writting music and music theory in general, one important thing to realize is that music theory is not a set of rigid rules that you must follow at all times. It is simply a set of guidelines that can help you understand what's going on in music and it can also provide certain formulas that sound good in general. However, if you write something that seems to break certain theoretical guidelines but you like the way it sounds then that's ultimately all that matters.
So to answer your question, if you really like the way it sounds then you can use it. Period. :)
From a purely theoretical perspective, you are also on solid ground here as well. (I IV V vi IV) Each chord leads nicely to the other one. You could even add another G at the end of it because the V chord leads back to the I chord very nicely (I IV V vi IV V), but it's not necessary. Try both and see which one fits best for your song.
Yes, you certainly can combine them. What's more, you are about to make a big discovery that can at least double the number of chord progressions you know, in one fell swoop.
That A minor is what's known as a substitute for the C major chord. Any time you see C major and feel like a change, try A minor and it's likely to sound good. This is called substituting the relative minor of a major chord.
It also works in reverse: you can substitute a major chord for its relative minor.
But how do you find the relative minor? Just count down three semitones. C to B is one semitone; B to B flat is another semitone and B flat to A makes the third semitone.
So, taking your chord progression, you can substitute D minor for the F major and E minor for the G major. When you do the math you'll find that your choices for chord progressions have just gone through the roof.
Starting with C, the following sequences are available:
C Am F G
C F G Am
C Am G F
C F Am G
C G F Am
C G Am F
That's six ideas, without repeating any chord.
Most songs will have many more chord changes than that, so hopefully it becomes apparent that the number of sequences grows exponentially. One of the reasons so many songs have been written with very few chords. Obviously, once notes and their durations are put into the equation, it becomes even more apparent. More tunes than the stars in the sky, possibly?
Music is about melodic and harmonic movement, and what you choose will depend on the effect that you want. The standard chord progressions work so well because the movement from chord to chord pushes forward until it ends and relaxes at the final cadence. When choosing your own chord progression, listen to how the chords interact with each other. Does the progression seem to intensify and gain energy, or does it relax or even seem like a letdown? This will give you a clue what chords you want, even if they do not follow a standard progression. You don't need to follow the "rules", but you do want to really listen to what your progression is doing in terms of push/pull, tension/relaxation in the music.