To figure out the best time to plant a shrub, it's necessary to understand a plant's growth cycle--what it does in each different season--and to know the climate and soil type where you are planting. Because gardening practice varies with local conditions, it's difficult to make general rules that apply everywhere in the country to every kind of shrub in every kind of soil. That said, the best times to plant a shrub are early fall and early spring. Here's why.
For most shrubs in most places, early fall is the best time to plant. This is because the huge, energy-guzzling growth spurt of spring and the stress of hot and sometimes dry summer days are past. The weather is starting to get cool. Transplanting puts a lot of stress on a plant because it has to grow new roots. In fall, growing new roots can be its only task. It doesn't have to make leaves or flowers or cope with hot weather at the same time, so it has more energy for root-growing and can get well established before going dormant for winter. As soon as growth resumes in spring, that shrub is ready.
Exceptions to the "fall is best" rule are for shrubs planted in heavy clay soil, because they may get heaved out of the ground by frost before they've been able to grow enough new roots to be securely anchored in the ground. Also, plants that are winter-sensitive will do better if given a long time to get established in their new spot before they face the cold.
Late winter to early spring is the next best time for transplanting. The shrub is still dormant or just breaking dormancy at this time. Plant as soon as possible after frost has left the ground to give the shrub the maximum amount of time to grow new roots before the spring growth frenzy takes hold and the shrub starts work on growing its above-ground parts also. In general, all plants are primed to grow in spring, so they're likely to take hold well at this time.
Transplanting is actually possible any time of year except when the ground is frozen, though it causes more stress to the plant at other times than fall and spring. In hot weather, you may need to water the new shrub every single day, or even more than once a day, to make sure it gets enough water. This is because until it grows new roots into the soil in its new spot, the shrub can absorb water only from the roots in the bits of soil left over from before it was transplanted. In hot weather, the shrub can easily die from drought. Also, planting in winter when the shrub is dormant means it won't start growing new roots at all until spring.