I'm posting this as an answer so I can give more detail in case anyone stumbles upon this in the future.
Setting up the MOSQUITTO MQTT Server in Ubuntu 18.04 is actually not hard, but the steps are important.
Step 1: Install Mosquitto Software
sudo apt-add-repository ppa:mosquitto-dev/mosquitto-ppa
sudo apt-get update
sudo apt-get install mosquitto
Step 2: Open Port 1883 and start firewall
sudo ufw allow 1883
sudo ufw enable
Step 3: Verify Mosquitto is not already running
[Note, if any number shows, that is the PID of an already running Mosquitto. You can just kill it. Also, you can try: sudo service mosquitto stop]
Step 4: Start Mosquitto with verbose option
[Note: This starts Mosquitto without using any config file. It echos connection and status information to the screen. Easiest for quick debugging.]
Step 5: Check connectivity using local host
Go to your client machine (in my case a Windows 10 laptop) and run the MQTT client, connecting to the local address of the Linux Mosquitto server (in my case 192.168.0.144). You should be able to connect. In fact, you can do this step before you even open the firewall, since this is all on the local network, the firewall rules are irrelevant at this point. Until next step which is...
Step 6: Check Connectivity using web tool
use either: www.yougetsignal.com/tools/open-ports/ or
[NOTE: You will not get an OPEN state UNLESS THE MOSQUITTO BROKER IS RUNNING]
Step 7: If Port Shows Closed When coming In from Internet (ie not localhost)
Here's where I got tripped up. In my case, I have a Verizon Modem that ALSO has a firewall (because it has a router). I have my own wireless router, a tp-link Archer C1200, that I have plugged into the Fios Modem/Router. I started by putting the port forwarding in the tp-link. But that firewall comes after the Fios firewall so I needed to go to the first wall and do the port forward there.
And this is the second thing that is tricky. All of the online how-to's said I should forward port 1883 to the local IP address of my Linux Server, which in my case was 192.168.0.144. But that was not correct in my case. The Archer C1200 was actually the device that I needed to forward to -- it handled the correct distribution from there. It had an address of 192.168.0.152 assigned to it from the Verizon router. I still have both forwardings in place (ie the Fios and the tp-link) and my guess is that I need them both.
Now all pathways are open, you can follow the other Mosquitto instructions regarding logging, config files, Daemons, etc.
Hope this saves someone some time down the road!