Play F – G – A – B♭ – C – D – E and back to the first note again, and you have the scale of F major. With 1♭ in the signature (B♭), it is easy to read once you know the signature. Every chord in the table above is made only from these notes.
In F major the chords F, B♭ and C carry the weight: they are major, they are built on the strongest degrees of the scale, and moving between them sounds like a sentence with a beginning, a middle and an end. The minor chords ii, iii and vi share two notes each with those, which makes them easy substitutes.
Starting the same scale from D gives D minor, the relative key. Borrowing a chord from F minor, the parallel key, is a classic way to add colour: a minor chord in a major key, or a major chord in a minor one, for a moment.
The dominant seventh chord, C7, is the strongest way to lead back to F: it contains the leading note E, a half step below the home note, and resolving it feels like coming home. Nearby keys on the circle are C major and B flat major.
If the barre chords of F major are hard for you, use a capo on fret 5: the shapes C, Dm, Em, F, G, Am then sound in the right key.
Practise the key in three steps: the scale up and down, the seven chords in order, then one of the progressions above until the changes feel automatic. Ten minutes is enough.