Just wanted to check in to thank John and DARTrider for their great tips. We were able to get the whole shebang up and running this morning. I also wanted to thank my awesome wife Bayleh for helping me figure out what to do and how to do it.
I'm going to list our steps here, as much as I can remember them, in case someone else wants to do the same thing:
I installed a fresh copy of TRS 2006 on the new PC, from the disks.
I installed the service pack (SP1 which will bring the system to Version 3.0.9.2}, aka the patch, on the new PC.
https://www.gamepressure.com/download.asp?ID=12130
I opened the Content Manager on my old PC (beware! you apparently can't have them both open at the same time, on both PC's).
I clicked on the Assets bar, which is blue, above the list of Content, to change it to Text View so I could see a list.
Starting at the top of the list, I pressed Control and clicked on each asset that had a green triangle next to it. I didn't do all of them at once, or even too many of them, because if you do, Control Manager will choke as you're trying to save, and you'll get nothing. You'll end up making your first collection of assets and turning them into one CDP file.
Once I had my first batch of assets clicked in this way, I right-clicked on one of them and chose "save to CDP," which created my first batch of assets, which I named CDP1. (I made sure to keep track of where I was on the list so that I could continue on with Control-clicking for the next batch. If you don't do this and the program chokes, you'll have trouble figuring out where you last stopped).
My next batch was CDP2, and so on. I ended up with 30 CDP files. (30 batches of CDP's).
I put those 30 CDP files in a folder and used a terabyte hard drive to transfer them over to my new PC.
I closed the Content Manager on the old PC.
I opened the Content Manager on the new PC.
I clicked "File" and then "Import CDP Files."
I directed it to where I had moved the 30 new CDP files and clicked to import each one.
I right-clicked on one asset in the resulting list (it doesn't matter which one) and selected "Select All." They'll all turn blue.
I right-clicked in the blue field (it doesn't matter where) and looked at the drop-down menu. I hovered over "Edit," and then selected "Commit."
I started up Trainz and on the entry menu I clicked "Options." For Display Settings, I chose 1280x1024. I could have chosen much higher, but I figured it wouldn't make any difference if I did.
For Tuning Settings, things were already pretty high, but I increased the Draw Distance. For Advanced Options, I increased the Resource Cache Size to 512, the highest offered.
Presto.