The diatonic chords in a key are formed by stacking up thirds on top of a scale in that key. For example, the C scale is made with the notes C, D, E, F, G, A, B, C. The key of C major has no sharps or flats.
If you stack up two thirds on top of the notes of the C scale you will have the series of chords C, Dm, Em, F, G, Am, B°, C. The chords built on top of the first, fourth, and fifth degree of the scale are always major. The chords built on top of the second, third, and sixth notes in the scale are always minor. That's why we write the Roman numerals for those chords with lower case numbers, and tack on an "m" after the root note in the chord symbol. The vii° chord is always diminished.
Restricting yourself to choosing diatonic chords of a key will reinforce the feeling of that key.
1. Make a list of the diatonic chords in the key of A major, then check your work.
2. Make a list of the diatonic chords in the key of F major, then check your work.
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Diatonic chords of A major: A (I), Bm (ii), C#m (iii), D (IV), E (V), F#m (vi), G#dim (vii°), A (I).
Diatonic chords of F major: F (I), Gm (ii), Am (iii), Bb (IV), C (V), Dm (vi), Edim (vii°), Bb (I).