Probably 1993, when I first went to college. Very slow in those days.
Today, I have broadband. And due to the wi-fi network in my house I'm writing this on a laptop whilst slouched in front of the TV in my living room. Technology, gotta love it!!
Online to what? To the Internet, it was about 1990/1991 when you needed a heap of Unix commands to ftp something from a host.
About the same time, I got a Compuserve account.
But I had an email address and connection to BT Gold in 1987. I think that was through a 9600bps modem (although it might have been 14.4k). I remember it took the best part of a day to connect a modem to a PC though.
I first started using online services in 87, dial-up services like Data Star and Dialog which charged you per minute and for any content you printed or downloaded. First used with Internet at work in about 93/94 and started looking at NSC in 97. Didn't get online at home until 99.