Q: Should the chickens' feeders and waterers go inside the coop, or should they go outside in the run?
A: That's a good question! Chickens need to have access to their food and water at all times when they're awake. However, once they return to roost at night, they sleep soundly and won't get up to eat or drink. (The exception to this is if you have a broody hen; sometimes a broody will want to get up at night to eat and drink, rather than during the day.) But because your chickens won't, for the most part, be eating and drinking after dark, you can choose to have your food and water outdoors or in. It is done both ways.
My personal preference is to keep feeders and waterers inside, if the coop is large enough to allow it. The reason is that outdoor food can get rained on, and wet food can mold. Bad for your birds! Having feed outside can also attract predators or rodents... or sometimes it will just attract pests like squirrels and wild birds. You don't really want to waste money feeding the whole neighborhood of animals on chicken feed!
This is all personal preference, though. There are plenty of people who like to keep their feeders outdoors, and in fact many of the small starter coops you can buy don't have much room to place feeders and waterers inside. Instead, many who have small coops like that place the feeders and waterers in a sheltered area outside, within a secure run. This protects the feed from rain and from predators, both. Alternately, you can purchase very small feeders and waterers designed for a "microflock," which can be hung on the coop walls, leaving floor space open.
If you do have a preference, be sure to build or buy a coop that will accommodate where you'd like to place your feeders and waterers.