Ok, so I am late catching up, but I need some advice.
This will be my first Digital SLR and I will only consider Nikon or Canon. Am thinking of 10 to 12 mp, lightweight, 3" vari-angle screen, about 18 -- 200mm. Not too many bells and whistles. I am very sorry to be giving up my magnificent Nikon film camera, but one must move with the times. Any advice will be gratefully received. Rather like Nikon D5100.

Used/Refurbished or New? Like with any camera the body is less important than the lens. Do you have a price range in mind?

Go to a camera shop and have a play with them all. In addition to features, you need one that feels comfortable to hold & use in terms of size, weight, balance etc.
Dump the swivel screen and get the Nikon D7000 then you can use all your older autofocus lenses, or use them as manual focus on the D5100. On the D5100 and D3100 to make the body smaller has the focus gear in the lenses not the camera body, these cameras will not focus with older autofocus lenses, which are often these days much cheaper than the new lenses.

the fact of life.
what ever you buy you will hate -- but you had to try it anyway.
before you drop down your credit card, beg borrow and steal as many of your friends cameras you can. though the basics are the "same," you will find that there are things you like about one over the other.
canon v. nikon.
the big deal breaker of on system over the other for most people coming from film is... manually focusing.
nikon turns clockwise towards infinity and canon turns counter clockwise toward infinity.
number 2, off the top of my head, maybe... no canon lens offers on-lens apeture control ring (you may not even care)
if you have a good set of nikon glass, keep on a nikon system that supports it. low budget canon choice, used 5d mk1. that should go for about $900, great workhorses, full frame and a bag of chips.
Thanks guys, for all your good advice. Will try everything you have mentioned. Have a great set of Nikon lenses that I don't want to just throw away.