There's a lot of different ways to build chord progressions in a key, but since you are very new to this I suggest just sticking to building chords in the key at first and stick to basic triads or sevenths in the key. How you would do this is not that hard and I'll explain it for C major and A minor.
Typically chords are built in thirds (every other note) so if we were to build three note chords (triads) based on the C major scale C, D, E, F, G, A, B
we would have:
- C major (C, E, G)
- D minor (D, F, A)
- E minor (E, G, B)
- F major (F, A, C)
- G major (G, B, D)
- A minor (A, C, E)
- B diminished (B, D, F)
The A minor scale which contains the same set of notes A, B, C, D, E, F, G
and likewise has the same chords:
- A minor (A, C, E)
- B diminished (B, D, F)
- C major (C, E, G)
- D minor (D, F, A)
- E minor (E, G, B)
- F major (F, A, C)
- G major (G, B, D)
While building triads off these scales is the same the distinction is important as building chords of the C major scale implies that you are in the key of C major and building chords of the A minor scale implies that you are in the key of A minor. This becomes important when you want to establish a tonic (home note).
For now as a first step, just play with these chords, see what you like and what you don't. There is more thought you can put into these progressions, but just experimenting you should be able to hear some of them.
Once you get comfortable with the general idea of these chords you can start to focus more on functional harmony and building progressions with the function of each chord in mind. This answer goes into much more detail about this in major and minor keys:
Guide-lines for creating a simple chord-progression?