Lilacs do not require too much Pruning - Here's how.
All that is normally required of a Syringa (Lilac), is the removal of faded flower. No more - unless you want to re-shape the bush. In the latter case, do it as you wish, right after flowering.
You may miss out on some flower the following year.
It is important to take off the old flower growth before it starts to set seed. This will then allow new shoots to form, which will be the flowering growths for the next year. The sooner you 'de-head' - the faded flowers, the better the following year's flowers will be.
Simply snip off the dead flower, just below the old flower stalk and above a pair of buds which will probably be starting to show though. A good clean cut with sharp secateurs will ensure quick healing and getting the new flowering growth moving.
All varieties of Lilac should be treated in this way. this includes the Syringa vulgaris, S. Hyacinthiflora types.
Basically it is just dead-heading the old flowers before they set seed.
Older lilac trees and bushes can be cut back quite hard, if the tree has started to get a bit top-heavy, or simply bare at the bottom. This is more successful with the Syringa vulgaris types. With other types, it is better to cut back just a third of the growth in each of three years. That is to say pruning one in three main brances in any given year.
Hard pruning should be done ideally in early Spring or late Winter. If the Lilac is cut back to within a few feet of the ground, it will soon start to send out new shoots, which can then either be thinned out as necessary to form a good framework, or simply left for the shrub to find its own form.
This hard pruning back will probably deprive the shrub of flowers for the following Spring, but will then continue to grow into a full bodied shrub with many new flowering shoots.
What are Suckers? I am thinking they are an insect / pest or something similar. Suckers are actually new shoots which grow from the base of a tree.
Some Lilac trees and bushes such as Syringa vulgaris, are prone to having suckers growing from the base. These will normally show through as having different type of foliage to the normal bush, and should be removed by breaking rather than cutting if possible. If the suckers are left to grow, then they will eventually take over - robbing the 'cultivated' variety of its vigour.
Species Lilacs such as Syringa alpina, Syringa pubescens, Syringa microphylla types can be treated as for the main group listed above.