I remember taking infrared pictures with a simple filter and black and white film. How can I do something like this with digital cameras? Or is this even possible?
Yes, but not well on most standard DSLRs. Most digital cameras have an infrared-blocking hot-mirror filter on the sensor to improve the image captures in the visible spectrum.
In order to do take infrared photos it is best to buy a camera designed for that purpose or modify a camera by removing the filter on the sensor.
There is a great overview of digital infrared photography here: Infrared Photography with a Digital Camera
The Sigma DSLR cameras do straight IR work without long exposures or altering the sensor - all you have to do is remove the dust protector (which takes a moment and requires no tools, and is easy to put back).
At that point you can do what is called a full-color IR where you capture the other portions of the spectrum but extend into IR, or you can use an IR-Cut filter like the Wratten filters (87b being an example of that). I have a Cokin-P filter holder, that I use with an IR-Cut Wratten gel - that way I can frame and focus and quickly drop the IR filter down into place before shooting.
You can find a number of images taken with an older version of the Sigma DSLR here:
You can achieve a similar effect without modding your camera by buying a Near-IR filter. These usually stop you waaaay down, so your exposure is 25–30 seconds, and you can't see anything through the viewfinder. Requires manual focus too. After taking the capture, you have to post-process the image — I forgot the details but it involved switching reds with blues in the channel mixer. Was way too much work for me, but I have seen some really successful shots with it.