I used to wake up in the middle of the night and sneak downstairs with my brother, lugging our family's first computer, a Compaq Presario all-in-one (similar to the gumdrop iMac) onto the dining room table so we could connect to the internet. With a 486DX2/66 and 4MB ram, we'd load up Tabworks and use the /worst/ browser in existence (don't recall the name, but it was probably a rip off of Mosaic) and dial out to IDT.net to connect to the internet.
Later on, after saving money to upgrade to a whopping 12MB ram and Windows 95, that same computer allowed to to play multiplayer Diablo, which was my first taste of IRC. Naturally it was all downhill from there, and I played Diablo for two years straight.
There were a lot of fights in our household over phone bills and busy dial tones, and I'm sorry that my sister was left stranded at school with no ride because I was busy downloading FreeBSD, but now I'm a successful software engineer and budding entrepreneur in the education space, and it's all thanks to those formative moments panicking at 3 am, trying so hard to muffle the sounds emanating from my 9600 kbaud modem as it connected me to an exciting new world.
It probably had a modem that didn't pay Hayes to license the patent on enforcing a delay in switching from data mode to command mode (+++ did that, and then ATH0 was the AT command to hang up the phone -- normally you'd have to wait a couple seconds, type +++, wait a couple more seconds, and then you'd get an OK and could type AT commands).
Oh man, I used to have the exact same phone line fights with my dad. A friend of mine had two phone lines at his house, and was the object of great and furious envy. I'd all but forgotten the days when an internet connection was a special treat.
I don't think I am nearly as old as you, my first touch on the Internet was on a 133 Mhz Compaq Pressario with 64 Mb of ram.
My dad finally got a second line installed as it was starting to get annoying that I was on line pretty much 24/7. Never did use Trumpet Winsock but the Windows 95 built in dialer still gives me nightmares.