One of the tricky things about pruning is that some plants bloom on old wood, and some on new. If you’re lucky, you held onto the plant tags and know the name of your plant and can look it up. But maybe you’ve lost that, or you’ve bought a home with established plants, and so you have no idea exactly what hydrangea or clematis you have. What you want to do then is just watch it for a season, and make a note of whether the flowers are appearing on new green shoots only, or on old wood only, or (less commonly, I think), on both.
If they bloom on new wood, like this Limelight hydrangea, pruning is simple — you just want to prune in late winter / early spring, before it’s started putting out shoots. That way, you can be sure not to lose flowers. Typically, you’d remove about a third of the plant — you can just trim it down by a third; you can also take out any criss-crossing branches (especially if they’re rubbing against each other, which can create a wound that makes the shrub vulnerable to disease).
It looks a little bare after pruning! But soon it will have a host of new green shoots, and after that, lots of panicle limey-white blooms. You can also prune for shape, although there’s only so much you can do with that — I planted this one a little too close to the house, so it’s sort of lop-sided, reaching for the light. Oops. Still pretty, though. 
If you can stand to leave the cuttings in a pile somewhere, rather than packing them away immediately, any little pollinators nesting in twigs and such will have a chance to wake up and come out. The cutting pile will also break down some over time, making mulch for your garden and enriching the soil. Bigger twigs and branches are great to save for backyard fire pits.