First of all, this will not be a do-it-yourself job.
If you can, connect the telephone line to the new computer. You can save a
little money on the DSL modem (although an external modem provided by your
DSL provider will be much preferred -- see below).
The old computer will need a network card. You can get a 4-port hub to put
between the two computers at Walmart for about $45. If you can't run the
cable indoors, get some CAT5E cable that's certified for outdoor use and run
it around your house under the soffet. For your setup, wireless may or may
not be cost-effective, but if it isn't configured correctly then your
neighbors will be using your DSL connection.
If you can't run the telephone line to the new computer, the old machine
will need a second network card and you'll have to buy (or lease, depending
upon your service provider) a DSL modem. DON'T get an internal DSL modem.
An external DSL modem should provide NAT (Network Address Translation)
which can prevent some kinds of criminal access to your machines.
Use Internet Connection Sharing to share the connection from the "host" PC
(which will have to connect to the phone line) to the "client" PC. You can
unplug the telephone line to the old PC.
Use some firewall software such as ZoneAlarm on at least the "server"
machine. While it's sort of okay, I wouldn't rely on the built-in Windows
XP firewall. (Note: Some folks abhor ZoneAlarm, but none of my clients with
home computers and DSL connections have ever complained about breakins.)
Your internal DSL modem probably will not provide even NAT, so it will be
important to use good firewall software and tighten security as much as you
can.
However, you should hire a qualified consultant to come to your home to
configure your systems. S/He might charge you a hundred bucks, but it has
to be done right. Make sure s/he'll do more than just plug in the machines
and configure the modem. If s/he says that there's a lot of junk on the
machines (especially the old one) that needs to be cleaned up, and if s/he
complains about the lack of a local security policy on Windows XP Home
Edition and s/he'll need a little more money, then s/he is a keeper.
However, when the consultant is getting ready to leave make sure that Client
for Microsoft Networks, File and Printer Sharing, and NetBIOS are disabled
on the network connection to the DSL modem. If they aren't, tell him or her
that the job isn't finished and you're wondering about his or her
qualifications and will need a discount.
Use Windows Update and keep your antivirus software updated religiously.
Set the machines to update automatically once you've got the DSL service up
and running. Keep your machines spyware-free by regularly using a tool such
as Ad-Aware.