Dropbox is supported by many other iPad apps, so it is almost an essential way to move files between devices. Click on the share button on an app and Dropbox will probably be offered as one of the options.
This app is a bit clunky but is useful if people already make use of Google Drive.
Allows you to connect to and synchronise with network drives. It worked well in my previous workplace and was the preferred way of storing documents because you could sync with network copies of documents thereby minimising the risk of ending up with multiple, slightly different, versions of the same thing. I could also use WebDAV to mount drives on my home computer, so this was an easy way to transfer stuff between my home computer and my iPad.
If your school doesn’t have WebDAV support (see above) this Remote Desktop client for the iPad may provide a (slightly awkward) way to access school network drives.
What other storage and transfer tools would you recommend? And, what is the best way to share documents with pupils, parents or fellow teachers?