The minimum system version for using a Classic Mac on the Net is generally agreed to be System 7, e.g 7.0.1*, 7.1, although there are some notes on getting a System 6 Mac on the Net at Matti Haveri's Internet FAQ. What you won't be able to do with System 6 is use the Mosaic or MacWeb browsers, but Fetch FTP client and an old version of Eudora email client (version 1.3.1) should work.
Almost any modem can be used, but you are unlikely to benefit from the increased speed of any modem over 28.8K, due to the serial port limitations of the Classic Mac.
I initially used a US Robotics Sportster Vi 14.4K to connect, and then replaced it with a US Robotics 28.8K Mac & Fax modem. I know also of someone connecting a Mac Classic to the Net with a Global Village Gold II 14.4K modem.
The good thing about using an older modem is that examples of such modems can generally be obtained for next to nothing, as they are seen to be too slow for todayís multimedia Macs and PCs. Check out your local computer store and second-hand street markets, or put a "wanted" notice on your notice board at school, college or work, and you should find one for very little cost. What you will need with most modems is the AC Adaptor to power it, and these can be quite hard to find if you don't get one supplied with the modem. A "Generic" AC Adaptor can be used, but you should ensure that it supplies the correct voltage and current. The USRobotics modems I have need 9.2v at 400mA, and will not stay connected powered by an adaptor supplying 300mA.
What you will (probably) need is the Modem Init String for the particular modem model you have, as this is used in the Connection Software to initialise the modem prior to connection. This can generally be found at the modem manufacturerís website, or by making a search on one of the major search engines on "Modem Init String".
Modem & Telephone Cables
On the "Front Page" of this website, you will see I make reference to a Hardware Handshake Cable which is required to connect your Mac to your modem. This resembles a standard serial connection cable, but the pin connections are different.
I wonít go into the technicalities of why this kind of cable is required (mainly because I donít fully understand!), suffice to say you need one! You should be able to pick one up from an Apple retailer for a reasonable price, they are not expensive. It may be described as a High Speed Modem Cable.
Regarding the telephone cable to connect your modem to the telephone socket, you will see on my Home Page that I said I had a problem with the wrong kind of cable. I think the cause of this was trying to use a US-produced modem with a UK cable. You will know if your cable is not suitable if the modem does not dial out (if the modem volume is turned off, you can check if it is dialling by putting a telephone socket "splitter" on your line and listening for the dialling on your telephone).