Rather than lift all of your fingers, keep your middle finger (#2) pressed down on the 2ndd fret of the D-string and rotate your hand around it. Using pivot fingers helps you make smooth chord changes by allowing you to move your hand as little as possible.
How long does it take for smooth chords to change?
most basic chords, the changes should come in a day or two depending on how hard you practice. some fingerings i still have problems with because i may use a different fingering that feels more comfortable for me. mostly though, it all boils down to practice.
Do chord changes get easier?
It does get easier. After time, you get over a certain point and it starts to get easier. when i very first started my mother showed me a few chords.