Step 1: Get yourself a crappy digital camera
Option (1) is certainly easier, and a number of instructables use that principle:
Infrared Ir Webcam
Take Infrared Pictures With Your Digital Camera
A better diy infrared filter
But there are some drawbacks to using an unmodified camera. Typically the exposure times are so long (1 second or so) that you need to use a tripod. Option (2) is more effective and you can take better IR pictures, if you're willing to take your camera apart.
Here's the catch. This project isn't difficult per se, but it involves handling a lot of small and fragile camera parts. There's a very real chance that one little slip up could turn your nice digicam into a very shiny brick. So don't try this on a camera unless you're willing to accept the risk of breaking it. I got this used VuPoint 3.1 MP camera at Ritz on the cheap. Used cameras also tend to be very inexpensive on eBay. If you have an expensive digital SLR you want to convert, you might consider paying a professional (like this place) to do it for you, which costs ~$300.
You will also need a lighting filter to block out all visible light except for red and longer wavelenghts. For this I am using "Congo Blue" (Lee #181 or Rosco #382) available from B&H for about $10 after shipping.