Well, I got it done! I'm now running the latest version of High Sierra. It took a little doing, though. First, I didn't have the original video card that came with the computer. I'd purchased it used and it came with a GeForce GTX 680. I thought it was the Mac Editin, but more on that later.
So so I found an original video card on eBay, bought it, and installed it. I also removed the three extra drives I had in the computer and disconnected all external drives. Then the firmware update for booting went fine and so did the install of High Sierra. I went with the 10.13.6 install because it had the proper firmware.
I then did some other software and security updates. But iTunes would still not recognize my new iPhone SE. However, I found a thread where someone else had the same problem. I downloaded an earlier version of iTunes, installed it, it searched for some new stuff, and when it was done, I now had an updated version of the latest iTunes and my new phone was recognized. I then restored the new iPhone with the backup from my old 5S and all was good. I also ran a new backup of my hard drive, using Carbon Copy Cloaner.
Now onto Mojave. As it turned out, my video card did not say Mac Edition on it. Rather than risk trying to flash it, I opted to buy yet another video card from eBay. It arrives on the first of July. Once I get that, I'll still need a new Bluetooth/network card. Then I can take the computer to Mojave. I'm hoping that will last me awhile.
And thanks to all for their help!