First thing you need to remember is that any disturbance of plant roots, regardless of the hardiness of the plant, will make them irritable. Your goal in all things is to keep the disturbance to a minimum. Remember that above all and the rest of the tips fall into place.
First and foremost, when I go to move a plant I will first dig the hole where the plant is going. Dig this hole deep and wide. I have a hand spade I use for transplanting and I will dig the hole about the depth of the spade and about twice the width. I would rather have the hole too big than too small.
Once the hole is dug, go over to your plant. My plants are moved when they are still seedlings. They tend to do better that way. For a seedling I will dig my spade in about an inch or so away from the plant and go straight down. Go way deep. You want to miss the roots. Scoop upwards so you are bringing the plant and dirt with it. If need be, place your other hand over the dirt coming up to insure it stays around the plant. Take your plant over to your new hole and place it in. Pull the dirt around it well, but try and keep the level of soil about the same as it was in its old home. Also would be a good idea to have the same side facing the sun as was facing the sun previously.
There is no way to move a plant and not disturb the roots at least some. What you need to do is protect the plants so they can have time to adjust their root system and recover from the roots being disturbed. There are a couple of ways you can do this: either cover the plant as soon as it is transplanted,( use straw, a planter pot, etc) or transplant on a cloudy day preferably in the evening as the sun goes down. If I disturb a plant too much I will do both. Once the plant is covered, let it stay that way for a few days. Check the plant after a few days and if it seems happy, you can uncover it. Uncover your plant in the evening around sunset to let it adjust to tomorrows sunlight gradually. The object here is to keep it out of direct sunlight until it has had a chance to recover from being moved. This is one I transplanted a few days ago. I checked it last night and uncovered it cause it seemed happy.
Water is essential when transplanting. As soon as you move a plant and get the dirt around it, give it a good long drink before you cover it. That water will help settle the dirt around the roots as well as give it plenty to drink on while it is adjusting.
Now I need to tell you that you will lose some. It happens. With experience you will lose fewer, but there will still be a few who can't take the pressure! I always give my plants a stern talkin to when I move them. It goes like this " I know this isn't your first choice, but I am giving you the best chance you have. It is up to you to take it or leave it". I have to say the vast majority of my plants opt to take it. Very smart of them I must say.
For the very adventurous gardener, you may one day try separating multiple plants that have intertwined their roots. I did that this year with cucumber plants. The only way I could find them in the store was in one pot and about 4 or 5 growing together. I took them out of the pots and submerged the roots in water washing away the potting soil. The water also allowed the roots to untangle themselves. Very slowly and gently I moved them apart, replanted them in pots with potting soil and kept them in the house a few days. Once they had recovered from that (after about a week or so) I transplanted them to the garden. I lost one of the 5 I had, but since I only wanted 4 I was out nothing. They are thriving now.
So there you go! An actually helpful post! Who knew!