Netscape? Pft. I started on the web with NCSA Mosaic. Of course, by the time it came out (January 1993), I already had an online presence for about 9 months. Of course, I don't count the time before that when I went on dial-up BBSes, since those didn't connect to the actual internet at the time.
As for our family's first computer, we had a TRS-80 Model 4. I don't remember when we got it, but it first came out in April 1983. That monster was actually still running up until ~2005!