Learn how to use edit mode in the Files section. This allows you to move, copy and rename local files and folders.
You can select files and folders by tapping on them.
Using the buttons on the bottom you can Delete (1), Move (2) and Zip (3) the selected files and folders. Those buttons become active as soon as at least one file is selected.
Tapping on the button (4) on the right side of each item, you can Rename and Copy the file or folder.
If you tapped on the Move button in the previous screen, Textastic will bring up a list of your folders. Choose the folder where you want to move the selected files or folders.
You can expand folders by tapping on them.
Once the target folder is selected, tap on the "Move" button in the lower right corner.
If you tapped on the button on the right side of a file or folder, you can choose to Rename or Copy it.