Do I have to access the files I put in SharePoint Online and OneDrive via my browser?

While you can always access the files you store in Office 365 from your favorite browser, another great way to interact with them is via Microsoft’s OneDrive sync client.  Syncing creates a local copy of your OneDrive for Business or SharePoint Online Document Libraries on your machine that you can access via Windows File Explorer.  In addition to allowing you to use your files offline, this gives you the ability to work with the files using the familiar File Explorer interface. 

Microsoft has great guides for getting started with the OneDrive sync client:

You can also access the files you store in Office 365 from your mobile device.  You can find more about these and other Office 365 mobile apps here.