Start by watering the plant you're going to move and trimming back the branches or top growth. I also think this is a great opportunity to prune out dead spots and old growth to help the plants rejuvenate once it's replanted. And it makes handling the plant easier.
Next, dig an ample-sized hole for the plant's new home. Set the soil aside and work a little pit of compost in there.
Using a sharp-edged shovel, cut a circle around the shrub or tree you're removing. Dig down at an angle to clean cut the roots. Once the plant is completely loosened, carefully lift it up. Try not to loosen the soil around the roots too much. When moving your plant to the new spot in the garden, you may want to place the plant in a pail, on some burlap or in a bag for the journey.
Once the plant is in the new hole, pack in the soil around the roots, carefully not to leave any air pockets. Finally, give it a good soaking. It's essential to water heartily throughout the first year to promote healthy roots. You'll want to wait a year or two before fertilizing. And hey, if you need any more help, the First Editions website has information on soil preparation along with planting tips.