USB-to-USB doesn't work these days unless you buy a clever cable and some clever software. The cheapest route, if both machines have a network card, is a Cat5 crossover cable.
A USB hard drive has longer-term potential.
If you're going to keep the old machine and use it for something else (eg a fileserver), then a modem/router and two Cat5 non-crossover cables may be the way to go.
BOAC's PMed advice, whatever it was, will be AOK.