Hi Bob
I'm Going to do something similar with 2 computers in the near future (maybe 3?)
I agree There are lots of Ways to Do it Lots of right and Wrong ways

So even I am going to be feeling my way about Semi blindly if I go ahead and upgrade to a Network

Shopping list : You'll need a Router (an ordinary DSL modem can only carry 1 channel) Price is about the same as as the Modem
Ethernet cards (cheap) These are needed to make the Network link on the Conputers
Network cables (prices vary wildly) But can get them for $3-4 each (rough guess)
As for the connection method it's very much trial & error if you are working from scratch
I post on another Site
which does have a forum section (where I don't post to) on this Subject . It's not an easy thing to explain in several postings ,and any of my "advice" is shakey at best

PS A Keyword search for
"Local Area Network" Will bring up countless sites on the subject

About 16 Million on Google alone
Suggest you start your search in the Wikipedia and Webopedia.com links (first links on the search above)