So, the main error I kept running into, (whether using the installer from my package manager, or from using Tor's instructions for using apt from the command line), was when attempting to launch the Tor Browser for the first time.
What is supposed to happen is something along the lines of, "go fetch the most recent version, and apply updates before the first launch". There could be other settings being made here too, I'm not sure. Either way, I would see some progress bars flashing by, then, a window would display "Error 404".
The solution for me was:
1. Remove Tor Browser. I did this using Synaptic, my favorite package manager. I only removed the "tor browser/installer" package, not Tor itself.
2. From https://www.torproject.org/download/ click on "Download for Linux". This will download a .tar.xz file to your download location. By default, this should be your "Downloads" folder.
3. Open a shell/command line interface and navigate to the folder where the "tor-browser...tar.xz" file downloaded. For example, type "cd Downloads" and press Enter.
4. To unzip the file, type:
sudo tar -xvJf tor-browser
The press the Tab key, to complete the file name. Press Enter.
5. After the files and folders have been extracted, let's move them to a proper location, instead of leaving them in the Downloads folder. Do this by typing:
sudo mv tor-browser /use/local/share/
6. Let's now navigate to that directory by typing:
cd /usr/local/share/tor-browser
7. Register the Tor Browser as a desktop application by typing:
./start-tor-browser.desktop --register-app
8. Tor Browser should now appear as an icon in your Applications Menu. Launch it, and the browser should launch without issue. Good luck!
Persistence pays!