1. Number one on my list is leaving burlap, straps, ropes, or wire baskets on freshly planted trees. In the old days, we thought some of these items could be left on the tree roots, but over time we are seeing more and more trees have problems and even die from leaving them on the root balls of trees. Please, remove these items.
2. A second mistake is what I like to call, "soil amendment gone crazy"! Adding too much soil amendment can be painfully bad for plants. Why? Because soil amendments can act like a sponge and can remain either too wet or too dry if applied in a concentrated form. However, you don't want to get cheap and skimpy with soil amendment either. The organic advantage that soil amendment adds to our sandy/clay soils here in Mississippi is great for the plants to achieve healthy growth. I generally till up my new shrub beds once, add my soil amendment on top of the existing native soil (about 1-2" deep, finely ground up pine bark), and then I till the ground once more. This mixes the soil amendment consistently within the native soil and provides a great organic mix in which those happy, new plants can grow.
3. A common mistake that even the pros make, which is a slow killer of plants, is planting trees and shrubs too deep. This will cause extended problems that will leave everyone wondering what happened. It's a slow and silent death that sometimes makes one question the quality of their soil. This is especially true when they see that other plants within their landscape are dying also. I have seen plenty of professional landscape companies get into trouble by not following this one simple guideline. You must look for the flare of the tree in the hole you are planting, and make sure that the flare is level with the surrounding soil before you back-fill the hole. Also, make sure that the base of the hole is firm and will not settle once you add water.
4. A fourth mistake is failing to consider the future size of the plant. Will he grow up to be big, tall, and handsome? Or will he be short and skinny all his life? In Florida, early in my career, I was trained to plant close together because hotel resorts wanted to instantly achieve full shrub and flower beds. Thank you, Disney, for setting that standard. I had to unlearn this training later. When planting correctly, one should consider that in 3 - 5 years plants will be approaching mature size and will need room to grow without being elbowed out by another plant. Something else you might want to consider is that planting too close to the house makes painting difficult and makes it hard to wash windows in the spring (you don't do windows either?).
5. Lastly, you know those tags that come on the plants when you purchase them? You need to do three, no, four things with them. Read them, follow them, remove them, and save one for future needs. If you don't remove the tags, they may actually twist tightly around the limb or trunk of the plant and cut off the food supply. We call this the "girdle effect" on plants. Also if they are white, the tags tend to look like loose litter in the yard. So keep 'em in a place where you can look at them later.