How can I remove Unity in Ubuntu and replace it with the Trinity Desktop Environment?

Yes, instead of starting with Ubuntu start with something like Lubuntu or Xubuntu so that you aren't (basically) forced to keep parts of Unity around for things to work.

Also, I wouldn't log in a GUI if it's going to get purged, instead use Ctrl-Alt-F1 (or through Ctrl-Alt-F6) to get into a console / terminal-only login from which you can purge GUI-related stuff and install Trinity without worrying about what happens to your logged-in session