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Styles are like symbols. If you change a style definition, you can update all instances of that style on the canvas. So one way to change color quickly across multiple objects would be to select an object with a style attached to it, change its fill color, and then click the Redefine Style button in the Properties panel. All the instances of that style will be changed to reflect the new fill color.
But there are other ways to change the fill or stroke colors of multiple objects. Simply selecting multiple objects on the canvas or in the Layers panel and then choosing the desired fill or stroke color will work. On the canvas, Shift-click or click-and-drag to select multiple objects using the Pointer tool (or the Subselection tool if an item belongs to a group). In the Layers panel, selecting a layer/folder will select all the objects in that layer; Shift-click or Command-click can also be used to select multiple objects or layers.
The one caveat to the above suggestions are Symbols. Symbol instances are easy to update in and of themselves by editing the Symbol, but they're somewhat separated from other items on the canvas. For example, you cannot directly change their fill or stroke color on the canvas, and styles contained within Symbols won't automatically update themselves the way they will for other objects.