First of all there is more than one way to finger each chord so commit to one and learn that smoothly before moving on to others.
If you are a beginner you need to know that it takes a lot of time to build muscle memory and good technique. And that is what you should focus on, good technique.
The chord form on the accepted answer is not the best, nor is it the only version of an open string G maj chord. Also, no offense, the pic of the hand position is terrible. This is not a good way to hold the guitar neck.
If you are fingering the C chord (x, 3, 2, 0, 1, 0) where numbers here are fingers (which also happen to match fret number in this case) then an easy transition to G maj would be (3, 2, 0, 0, 0, 4) on frets (3, 2, 0, 0, 0, 3). When playing the C chord the pinky should already be floating over the 3rd fret. All you need to do is move the two fingers (3, 2) from the (A, D) strings to the (E, A) strings, then lift the index finger from the C chord and place the pinky down. Even though it seems easy it still takes time to learn to move the two fingers in sync.
A perhaps little known concept is that many chords (especially those used in a key for progressions) have similar fingerings. There will likely be a fixed finger, about which the others move, or there is a common form that just gets moved around.
The pair of fingers (3, 2) placed in the 3rd and 2nd frets of consecutive strings are part of 5 different chords in open position. A list is provided below.
(3, 2) on (E, A) --> G maj (3, 2, 0, 0, 0, 4)
(3, 2) on (A, D) --> C maj (x, 3, 2, 0, 1, 0)
(3, 2) on (D, G) --> F maj (x, x, 3, 2, 1, 1)
(3, 2) on (G, B) --> Bb min (x, x, x, 3, 2, 1)
(3, 2) on (B, e) --> D maj (x, x, 0, 1, 3, 2)
Once you see that pattern I'd recommend just practicing moving the (3, 2) pair across the strings to different chords, like C --> F --> G --> C, etc.
Once this is in the muscle memory put the other finger in place. The hand should be relaxed, thumb behind the neck, palm NOT touching the neck.
When playing a G7 chord in a C maj progression life is even easier since the fingering for G7 is (3, 2, 0, 0, 0, 1). This is the exact same shape as the C maj chord just "opened up", moving (3, 2) down a string and 1 up a string (down/up in terms of pitch).
In this manner the entire I --> VI --> V7 group in C maj has a single fingering with (3, 2, 1) in a diagonal shape for all.
As for moving quickly you will need to be capable of forming the chord correctly first. Then I'd recommend the following, give yourself 4 beats on a metronome as a slow tempo. Play chord 1 on the first beat of a group of 4 (4/4 time). Allow yourself the next three beats to get to the next chord comfortably and prepare. Then on beat 1 of the next measure play chord 2.
I teach my students something like this...
On beats (1, 2, 3, 4) do the following (Play, mute, move to next chord, nothing), repeat
It may seem pedantic but it works, and gets results fast. Once you can play the chords back and forth don't speed up the metronome, but take beats away from your transition time and put it on your chord time. So, play chord 1 for 2 beats, then release, move, and set in 2 beats. Then play chord for 3 beats, and execute the other tasks in 1. After a few rounds you will find that you can execute the movement to the next chord in a fraction of a beat (the goal).