In just about every single recipe I write – with the possible exceptions of things that require refrigeration/freezing instead of baking – and almost all of those that I read, the instructions call for you to “cool on a wire rack” after your baked good is pulled from the oven.
Cooling racks really are an essential piece of equipment for a baker. They create a perfect place for a hot pan, or a de-panned baked good, to cool down evenly and quickly. Unlike a trivet, which will simply hold a hot object off of your countertop or table, the design of a cooling rack allows air to circulate through it, reaching all sides of a pan. As the pan cools down far more quickly than it otherwise would (especially if placed directly on a insulating surface, like many countertops), your cookies/cakes/etc. are prevented from overbaking in the residual heat of the pan. The same applies to baked goods that have been removed from their pans to cool down individually on wire racks. In these cases, racks also serve the additional purpose of preventing condensation, from the steam released of the hot food, from collecting and making the edges of your baked goods soft or mushy as your baked goods cool.
Size does matter, to an extent, when it comes to cooling racks. Cookies, cupcakes, muffins and other smallish baked goods don’t need a rack of a particular size to cool because they are small and can be placed close together on a rack, although you might need an extra one or two if you routinely bake large batches and have dozens of cookies cooling at once. Once you get into cakes, you’re going to want a cooling rack that is large enough to comfortably hold an entire cake. 10-inches seems to be the largest standard pan size (for bundt cakes and round cakes), so a rack that is at least 10-inches across should serve just about all of your needs.
Most of my racks are Wilton wire racks, in the 10″x16″ size, though there are larger racks (14″x20″) available which might be a better choice if you have a larger area for storage. If space is at a premium, a multi-tiered rack might be the best choice, so you can cool multiple things at once without taking up a lot of counterspace.