The short answer to making chord-changer quicker is practice. But there are some ways to make practice more fruitful. In this first set of examples we will practice moving each finger individually. The chart below shows each fingers' location in the chords A and D.

In Example 1 only the 1st finger moves from string 3 to string 4 (staying in the 2nd fret). To be thorough we then move the 2nd finger from string 1 to string 3 (also staying in the 2nd fret) and then the 3rd finger from string 2's 3rd fret to string 2's 2nd fret.

Once each finger can independantly make these changes it is time to move in pairs as seen in Example 2. Since both fingers are staying in the 2nd fret this should be fairly easy.

Finally, in Example 3 all three fingers are moving as a unit. It is perfectly normal if fingers 1 and 2 find their locations before finger 3 slides down to the 2nd fret. With a little bit of practice things will speed up considerably.