How do I get my Python scripts onto an Ubuntu server?

SCP is a reasonable place to start since you already have SSH working. It can copy individual files or recursively transfer a folder, and it can preserve timestamps too.

For a graphical option on Windows, WinSCP lets you connect over SSH and move files without doing everything at the command line.

Git clone is another option for the scripts if you keep them in a repository. You can still use SCP for the data files if those aren't in the repo. No need to set up a separate FTP server just to get started.

Getting the files over is only one part of running client work on a server. If Linux administration is new to you, consider paying someone experienced to help with the setup. Keep regular backups, and make sure you understand commands before running them.