Hi; So I'm, a very new beginner.  So like others, the A chord was a tight fit with 3 fingers jammed on 3 strings.

What I figured out;

1= make the E chord. 1finger on 1st fret, G string

2= 2nd 3rd finger on the A, D strings 2nd fret.  (So far so good)

3= Make the A chord this way.

4= 1st finger stays on the 1st fret, G string.  Keep it there.

5= 2nd finger bars the D, G string second fret.  (It can even touch the B string because the ring finger will press down on the B string).

6= Ring finger on the B string.  This finger cancels out the 1st finger on the 1st fret and any buz from the 2nd finger touching.

7= So I find this way I only have 2 fingers on the 3 strings and you can get away with being a little sloppy.

8= While playing the A chord, your 1st finger is now free so move it to the next needed string in the chord progression.  

     Have it ready as a pivot finger for the D, C, G chords.

I'm still slow but my chords ring out clean now.  Just working on chord changes time, not shapes.  Very happy.

PS  If this is common knowledge, let me know lol.  Being new and making discoveries is pretty fun

BTW I use this same idea a little when I make a D chord.

Cheers all.