Complete list of Android APK App Stores
Downloading APK files alone is easy but if you're grabbing just standalone APK's there is some limits to them, such as the fact you'll rarely get notified of updates to the apps you have installed. Given that there is a ton of Android app stores out there that all have their own APK for installation, it's more reasonable to pick you like and use that. Here's a list of all the Android App Stores available.
Keep in mind not all of them may be accessible from your area but this are the more popular stores with vetted content.