How to switch between chords
After learning guitar chords and the correct fingerings you should begin exercising in moving efficiently from one chord to another. The progress will be slow in the beginning, but after a while you get better at it.
Economy in movements
The key to moving fast from one chord to another is economy in the movements. The critical thing about switching between chords is avoiding unnecessary movements. If not necessary you shouldn't lift your whole hand from the fretboard but only move the involved fingers to new positions. And if some fingers shares the same positions in the "leaving-chord" and "the-going-to-chord" you should of course not move them at all.
A helpfull method is to think of your fingers as clusters instead of single entities. In other words: try to move two or three fingers in a group and put them down simultaneously. This is critical for being able to switch fast between chords.
Be sure that you use the right fingerings and notice that depending on chord sequences the fingering can shift. The ideal fingerings are what makes the switch to the next chord the most economical.
Then it comes to movable chords, such as barre chords, you should, in general, strive for small movements once again. For accomplish this you may have to learn two different chord shapes, that lets you use both the fifth and the sixth string (which are the most typical strings for the bass note in a chord) for the bass notes.
There’s not very much more to say on this subject – it’s not theory that make you succeed in this area but practice. As you're practicing different chord changes, you will by time be able to move easily between chords.
Here are some easy chord changes:
C - Am
E - Am
G - Em
More tips about chord changes plus exercises.
Follow chord changes in a song
To hear when the chord changes in a song can be difficult, and even harder is it to hear which chords are involved. To figure out the chords in a song you must have a pretty good “music ear” and that is something that takes time to develop. The best way to train at this is to listen carefully at songs and try to imitate what you hear.