There are a couple of different ways to fret the G Major chord on guitar. This particular fingering may allow easier changes between C and G.
Once fretted, play each string individually to make sure your fingers aren't touching a string they aren't supposed to and so causing buzzing or muting.

G Major is made up of the three notes G, B, and D.
Practice strumming the G chord and sing the individual notes of the chord.
Work on getting clean sounding chords and an even strum.

When playing in the key of C, G Major is the chord that is built upon the fifth degree of the scale and is known as the V chord and the dominant chord. The V chord has a tendency to want to resolve to the I chord so alternating between C and G or the I and V chords is a very common chord progression.
Practice switching between C and G. Play 2 bars of C followed by 2 bars of G and repeat.
Sing the root notes of each chord and learn the sound of the I - V progression.