GrapheneOS has some built in apps, namely for SMS, gallery viewer, camera, PDF reader, calculator, contacts, files, phone and web browser (vanadium, based on chromium). GrapheneOS offers no cloud. You are responsible for using the service of your choice to manage and backup your data. It is currently undergoing a transition for backup management, but otherwise you can make use of a selfhosted service like nextcloud.
GrapheneOS does come preinstalled with its own app store but that it is reserved to GrapheneOS apps and the distribution of certain google services which can be optionally installed using their sandbox. Besides that, you can indeed install the aurora store to get access to the free apps on the google play store, or actually use the google play store. They can all be installed and used simultaneously. Though you might want to be mindful of you install an app on one store to not update it on another as the two versions could work differently (e.g. an app installed on f-droid might have a different notification system than one on the google play store). You do not need to use nextcloud if you don't want to. GrapheneOS has no dependencies on any other additional app. It is a standalone OS. Once you install it, you use it however you want.
Edit: one key advantage of GrapheneOS is the possibility of using multiple users. You can (and I recommend it) separate apps into different user profiles. You can for instance dedicate one user profile to apps requiring Google services, let's call it Gapps. GrapheneOS then allows you to then pipe your notifications between user accounts, so if you are in your main user profile you can get notifications from apps running in Gapps in the background. Very convenient.