SLiRP is free. TIA is not free, but there is a free two-week trial period and it is inexpensive.
You can learn more about TIA at <URL:http://marketplace.com/tia/tiahome.html>.
More information on SLiRP is available at <URL:http://blitzen.canberra.edu.au/~danjo/>.
If you have a Macintosh, check out the Macintosh TIA Users' FAQ, <URL:http://seds.lpl.arizona.edu/billa/tia/faq.html>, for additional help.
"So what do I run on my machine at home?"
Exactly the same software you would use for real SLIP; as far as your PC is concerned, it is a SLIP connection. If you're unfamiliar with SLIP please check out a newsgroup relevant to your particular type of machine (Windows, Mac, or even Unix-based).