The prime reason to prune these shrubs is to keep them in reasonable shape for your landscape as well as to create well shaped shrubs. This is considerably easier if you have made wise choices for your garden by picking shrubs that are naturally an appropriate size for the space. If you want a neat three foot hedge, do not pick a tall growing shrub like forsythia!
Pruning is also important for keeping some multi-stemmed shrubs, such as winter honeysuckles (Lonicera fragrantissima), healthy.
For both these varieties, the first burst of blossom is still going to be on the old wood and occur in spring. When the bloom is spent, that is a good time to prune. Take the dead flowers off the shrub and cut the stems down to a junction. Check the shape of the shrub and prune back any lop-sided areas and small branches that are sticking out. Do not prune excessively. Over the summer, new branches will form and create buds for next year, or, in the case of summer blooming azaleas, the new branches will flower. Then leave it alone until next year. Making azaleas into tightly pruned boxes by constantly pruning every little branch that dares to stick out is going to make much more work for the homeowner than necessary and reduce blooms for next year. Additionally constant, heavy pruning makes the azaleas look like gaudy pink cupcakes which are fine for a kids party, but not for a landscape.
For the majority of spring shrubs, this annual pruning is all that you need to do. Allow the new growth to develop buds for next year and put your pruning tools away!