For the administrators, there could be ways to migrate it automatically. As a user this is more difficult. If you can do the federated sharing, you can share a folder on the new setup to the user on your old setup. This folder on the new setup is then visible in your old and you can copy data into that. Advantage would be that the data is transferred directly between the servers (which have a rather fast connection) and you don’t have to download and reupload.
If that is not possible, perhaps ask your IT service if they can help you (in case you have many files).
For migrating you must backup and restore the database and files. Because you use the same server, you perhaps do not need every part.
If you only use a few users and a few shares in your old nextcloud perhaps you can only copy the nextcloud-data-dir from the old cloud to the new nextcloud-data-dir and then “rescan” the new database. In this case you must defined the users before copy manually and shares per user after copy. In this case you must not dump and restore the database.