NotEnoughChips,
On the computer end (of your project) use Hyperterminal initially to send/receive data from the Arduino. You can implement a fancy UI later.
To connect using Hyperterminal, you simply have to work out which COM port your Arduino is connected to. Use Device Manager in control panel to determine this.
Oh, and you also need to know the connection speed. You set this in your Arduino code.
Btw, I like RoyK's comments about the moderndevice.com gear. Their gear also comes in kit form to cut $ even more.