There is difference if you need to run as service or simply run it, because your question implies that simply running it would be enough.
To auto start a program/script
Use Startup Applications program in your Ubuntu.
Put the (executable) script somewhere and create a new entry that basically just point to the script.
Make sure that the script has a proper shebang:
#!/usr/bin/env python or similar.
Also make sure that the entry is enabled.
What this does is basically creates a new .desktop file in
~/.config/autostart. You can hack away at that desktop file to your heart's content, just dont remove the
X-MATE-Autostart-enabled=true (this line could be slightly different depending on your DE)
To create a proper service
Ubuntu 16 uses
systemd and you need to create a service for that.
Creating a systemd service is a bit more difficult, but there are plenty of good guides like:
However it is mostly used for background daemons, not indicators or other GUI things, so you should stick with Startup Applications.