Step 1: Get a Webcam
If you have a Windows computer, you probably don't have a webcam built-in, and as such will have to buy one. What you are looking for is a "USB webcam". They're sold all over the place, and you get what you pay for. You can pick up a cheap Logitech for 25 dollars, and I wouldn't recommend spending more than 30 dollars until you know you're really a keen webcam person. Look for a high framerate (30 fps or more) and at least a 640x480 resolution. You may also want a built-in microphone, which is a pretty standard feature. (It's hard to find a webcam without a microphone, actually, but if you're buying a really cheap one, do check!)
It's worthwhile going to a large computer store (CompUSA, Compusmart, Future Shop, Best Buy, or whatever's large and near you) to have a look, as they tend to have one of each webcam set up and connected to a computer, so you can compare image quality. Some of the more expensive models do fancy things, like tracking (the camera follows you around the room) and have nice zooms, which may be useful, depending on your intended use.
I've included a photo of the Creative WebCam Vista Plus, which is a really cheap, very basic webcam, which is Mac and PC compatible. It's one I'd recommend for people just beginning to mess around with webcams, and you can find it and other Creative webcams for $15-20 USD if you shop around.
Now hook it up to your computer with the included instructions (most of them are plug-in-and-use now) and move on to step 2!
Step 2: Get to Chatting!
Let's say you want to talk to your friend, who also has a webcam, on a Mac. There are a few options, the easiest of which to use is probably iChat, a program which both of you already have. All you need is a .Mac account and you're set up automatically. Easy!
Windows Live Messenger is your friend here. Free to download at get.live.com/messenger. It even allows full-screen video conferencing! You will both need a Windows Live account, but it's free, and you can get it at the URL I just gave. If you don't want to have to download things (and frankly, who does?), you may want to check out webmessenger.msn.com. It's an online version of Windows Live Messenger, so you'll still need that Windows Live account (which, by the way, you have if you have a Hotmail account). But you oughtn't need to download anything.
userplane.com is one place to look for chat rooms, and ivideochat.com is another. Be extra safe online, and never meet up with anyone in real life from any of these video chat places. I can't really offer much other advice, except "don't download dubious downloads" and use as much common sense as possible.