I had a fax number connected to a regular fax machine/printer, too. After I got a computer with a built-in fax modem (surprisingly, those come standard with some relatively cheap PCs), the set-up was pretty easy.
Take the phone line out of the fax machine and plug it into the modem's phone input that is on the back of the PC.
Use software like:
And it works relatively easily. When you start the program, it should be on yellow light for a while, and eventually turn green (on "receive"). As long as the program is running and the PC is on, it should receive (barring phone/power outage). When it receives a fax, it saves it as an image in a folder of your choosing. You can then open it, print it, archive it, whatever. The important part is that it doesn't break down like a fax machine, doesn't need paper and doesn't use ink/toner. It's cheaper and easier to maintain. Obviously, you can still print any faxes you need physically.
Sending is just as easy. Type up a document, save it in a familiar format (I think .doc or .pdf will do), and select it to be sent. Cover pages can be programmed and sent automatically along with each fax.
Don't forget to buy a UPS to make sure a power surge or short-term blackout doesn't shut off your computer. Other than that, the most difficult part is finding a PC that has a fax modem.