(Also 36) It might not be my very first experience, but I remember computers in the school library being "locked down" such that you couldn't directly launch IE. But you could open Word and File - Open... "http://yahoo.com" (or whatever) which would launch IE. So at least some of my clearest early Internet memories are circumventing access controls.
I missed the AOL train, but I remember my cousins being into it when I'd visit them in "the city" (the also had cable television). I don't think we had a computer at my dad's house until Windows Me when I was nearing high school age.
I know my mom on the other-hand was rocking dial-up for email and browsing on a Windows 95/98? machine. But as things went, I spent a majority of time at dad's.