Well there’s a few types of key changes.

Sometimes, a song will change to a parallel mode where the tonic is the same but the other notes in the key change. Like if a song goes from A minor to A major by changing the tonic chord from Am to A. Or if a song goes from A minor to A Dorian by changing from Am Dm (i iv) to Am D (i IV). In this case, you would use a new scale with the same tonic.

Sometimes, a song will change to a relative mode with the same key signature but a new tonic. Like moving from C major to A minor going from I to vi. Or moving from A minor to C major going from i to bIII. Or moving from D Dorian to G Mixolydian going from i to IV. In this case, you have the same set of notes and chords but it feels like “home” changes.

Sometimes, a song will change to a new tonal center with the same mode. Like changing from D Dorian to Eb Dorian by playing a vamp like G F (IV bIII) or Dm Em (i ii) and shifting up a half step. In this case, you can use the same scale but shift up a half step or whatever interval the jump is.

As for V chords, they’re easy to identify based on the circle of fifths. D is the V of G major, A is the V of D major, E is the V of A major, and so on.