You will not get any more flowers this year. Lilacs (except for a few specially hybridized versions like Bloomerang), bloom for a few weeks in the spring and then they are done for the year. As soon as the blooms are finished, you should clip them all off. If you leave the spent flowers on to set seed the lilac will produce fewer flowers the following year. If you don't let it set seed, it will try again next year with a bumper crop.
If there is any pruning that needs to be done, now is the time to do it. Lilacs do not need a lot of pruning, but cutting out branches that cross and rub or minor shaping. By late summer, the shrub will be setting the buds for next year's flowers. If you cut those off, you are cutting off next year's flowers.
People talk about removing suckers from lilac, but honestly in the case of lilac, unlike trees, I'm not sure the difference between a sucker and a new stem. Lilac is not a tree (although it can be pruned and trained to look like one, called standard form - dwarf koreans are often done that way). Lilacs in nature are a large spreading shrub, that forms a colony by putting up new shoots.
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In the fall or early spring, you can plant stems pulled off the shrub. Dig down and pull the stem from as far down as you can, getting as much root as you can. Then just plant it and keep it moist, but not soggy, until your new shrub is growing well.