There's a thread about making music right at the top of this board. Could've asked that there.
Anyway, if you want to adhere to chiptune reality, you'll have to get software that kinda sorta emulates old school audio hardware. FamiTracker is an example of such software for the NES and is standalone. There are also plenty of VST plugins for you to use (in VST hosts) for that purpose, like basic64, quadraSID (both C64) and Triforce (NES).
If you don't want to be restricted and make what I heard was called "fakebit", just remember to use only basic waves in your synthesizers of choice (triangle, sine, saw and rectangle) and basic types of noise (white and red). And don't use effects, aside from filters.
Refer to this post
for slightly more information.

I quite fancy getting into making some chiptunes, and I wondered, short of wandering around second-hand shops and buying up all the old hardware I can find, if anyone knows of a good way to start making some on the computer. Free (or cheap) programs, equipment, what have you, that are a good start to the electronic music-making process. I can do real-life music making but have no experience on the digital side of things, so I would appreciate any experience you might all have had starting out digi-composing.