This is a simple how to guide moving files in Microsoft SharePoint Online site.
Login to the Sharepoint site, go to the folder where you want a file to move e.g. Shared Documents. In Library Tools, click Library. In Library, click "Open with Explorer" it will load the page wait for it.
You will see a similar Windows Explorer where you can cut and paste a files or folder. Keep in mind, you're doing this through the cloud, so the transfer speed depends on how big your files or folders.
Click the Browse tab to go back to the normal page of your sharepoint site.
The second option of moving files is using the "SharePoint Server Publishing Infrastructure". Here's how to activate this feature for your site.
In your SharePoint site, click Site Actions, Site Settings. In Site Collection Administration, click the Site collection features, then scroll down and look for SharePoint Server Publishing Infrastructure, click Activate.
Please note, when you click on Activate button wait for it to load. You will see a browser trying to load a new page. When it's done. Click Site Action, Site Settings. In Site Administration, click the Content and structure. This is where you can select a file or folder, click Action, then select Move (or Copy or Delete).
I hope this help you manage your SharePoint Online site.