There are numerous ways to migrate from one cloud storage provider to another. You can manually download all of your files from your current storage provider then manually upload them to the new storage provider. However, this takes time and can be slow, depending on your internet connection speeds.
Depending on the two cloud storage providers that will be involved in the move, you can use services like Otixo, Mover or similar services. If you’re looking to move files from Dropbox to Google Drive or Microsoft OneDrive, you can make use of numerous IFTTT scripts to make the move.