Anchor Fingers for A, D and E
Using your 1st finger as an ‘anchor' will help you move between your chords smoothly and easily.
Drop the anchor to guide the way
Keeping your 1st finger on the third string (G) will help you guide all of your fingers to the next position when you change chords. You don't need to keep the pressure down, but leave it in contact with the string all the time.
Start with a change from A to E. Put your fingers down for the A (1.); then lift your 2nd and 3rd fingers, while leaving the 1st finger in its place. Slide the 1st finger back one fret to the 1st fret (2.), and then place your 2nd and 3rd fingers down in the correct frets to form an E chord (3.). After a few times changing between them I'm sure you will find that the anchor finger makes it a lot easier to make the change.
Once you are cool with that, have a go at using the same technique to change from A to D. This time the 1st finger will stay in the same place. Try it.
Once you are there with that, move onto changing between D and E, still using your 1st finger as an anchor.
You should aim to be able to make any changes between the chords A, D and E while keeping that first finger in contact with the string. It doesn't have to be hard, but don't let it lift from the strings... if on the change from D to A you want to keep your finger down, it won't make any difference as it's staying in the same place!
At this point I still recommend trying to get your fingers down one at a time, usually in finger order, starting with your 1st finger, then 2nd and then 3rd finger. I would suggest still placing the 1st finger down and then the 2nd, just for consistency. It wouldn't make much difference really if you want to put them down together or the other way round though, so don't get worried about it.
This concept can be used any time a finger needs to stay on a string. It only works with certain chord changes, but use it where you can.
Try and maintain your accuracy as best you can while doing this stuff. The next thing we're going to look at is developing speed, (getting your chord changes faster) where accuracy is less important, but for other things, try and keep it as correct as you can!
Remember too that at the start you are bound to make little mistakes. Be patient with yourself and enjoy the journey. Little mistakes will get corrected with time and practice, so don't beat yourself up if you have not mastered the guitar in the first week. It may take a little longer!
Try doing it a few times slowly now and get used to the feeling and then it's time to really get your chord changes moving as quickly as you can in my famous One Minute Changes exercise!