These sugar-heavy syrups sink because they're denser than the drink you're mixing them with (an effect that's used for visual effect in some cocktails). However they should mix fairly easily. With hot drinks, a warm cup and adding the syrup first then the hot coffee, milk, or whatever should get a normal quantity mixed in but a higher concentration might need a stir. That's how the coffee shops do it.
In iced coffee drinks, the syrup is often mixed with the hot espresso and stirred before the chilled ingredients are added. If there are no hot ingredients at all, you might find it easier to use a small whisk or a fork, rather than a spoon, to stir vigorously into some of the water before adding the rest and any ice. Or a cocktail shaker works well, even if there is ice.
Much of this is because syrups are runnier warmer, and almost stiff if they're cold enough.