OK, peeps. I spent a long time today with techs from Symantec.
For the person who said fairly early I would need to uninstall Norton - here’s what the Norton people do:
You go to configure Firewall, and click on Networking. You can enter a range of trusted computers (like all the ports on your router), or individual IPs.
As for my problem, still not quite solved, but I’m a chronic pain patient (in addition to being an old bag, not a geezer), 
and that Symantec session is sending me to bed.
Hopefully, I’ll feel well enough tomorrow to come back here and read over the stuff all you nice people have written. In the meantime, I’m at the stage where the old computer wants to use the (now disconnected) Laplink, and the new one wants to use the LAN. :smack: Oh, yeah. The new computer CAN now ping the old one successfully.
Questions that I have barely enough sentience left now to ask:
Do I need to drag My Documents - and anything else I want to share - into the Shared Documents folder? Or does having defined the other computer (in each of them) as “trusted” suffice?
How do I “collect” the information I want to transfer? I’m mainly thinking about the configuration of things like Firefox and Eudora.
As I said in the beginning - I’m a total noob WRT LANs. I want to get this set up so I can use the new computer exclusively, and turn the old one on once or twice a month to synch them. Last question for now, that I know I’ll need the answer to before I’m done:
I assume I’ll need software to synch them. What’s the best cheap (or, preferably, free) software for the purpose? Like I said, I want it working so that it’s no harder than running my AV, etc., and saving the latest status to system restore.