For a dial-up connection to the internet, your Mac functions the same as your PC did - the connection will stay open (and your phone bill will go up) until you disconnect it, or your ISP logs you off. Sleep mode won’t close your internet connection for you.
If you’re familiar with Unix (or just want to be), the O’Reilly book Learning Unix for Mac OS X is a good one. Everything in it’s done through the Terminal.