My fondest memories of being online predate having web access!
I grew up in a remote corner of Kansas. BBSs were the thing in the 80s and 90s, and maybe CompuServe. But everything was a long-distance (expensive!) phone call. I had various hacks to get Internet email. Via a wonky FidoNet gateway was the best for awhile; my system dialed up a BBS in the middle of the night (cheaper rates!), quickly exchanged mail, and hung up.
Then I got a UUCP feed. Similar concept, but then I could also get... USENET! (Not the binaries people think of now, but discussions.)
Eventually I got into Free Software: FreeBSD and Linux. I remember going to a computer lab in a college in summer when it was empty with a large stack of floppies. I'd download Debian installer disks via FTP on about 6 computers at once, write them to disks, and repeat. Woo!
Finally PPP became available in my area. Affordable Real Internet at home! I promptly put up a personal website (ISPs gave those out standard at the time), ran my UUCP stuff over the Internet, etc. I remember the thrill of being able to access news sites. From Kansas! For Free! And then there was RealAudio even. And maybe even RealVideo, if you were super lucky.
A lot of libraries were set up with telnet access to their card catalog - how cool was that, seeing what's there and even renewing my books from home!
Then I moved to a city and got myself a 128Kbps ISDN line. That was hopping! I could run a live SMTP server myself - no more UUCP. Somewhere around that time, Slashdot popped up as a popular site. I also used Mapquest to get -- and PRINT OUT -- driving directions. Then you started having search engines - AltaVista, Excite, etc. And Deja News - the Usenet indexer - was one of the best ways to find technical information. Sort of like a "site:reddit.com" search now.
When Amazon became a thing, that was cool. No more driving 40 miles to a bookstore!
Eventually I got a job at a University, and my desktop machine ran Linux. It had an un-firewalled public IP, so I promptly started hosting my Linux-related website on it.
Oh, and did I mention IRC? (Chat rooms)
But you'll note, I named the websites last. My fondest memories weren't really about websites. They were about communication and community. I never really liked web forums ("I can use my own client for email and Usenet, and get all of the groups in one place; why should I register 20 different accounts for forums that will go down with the server admin gets bored of the project?").
Actually, they still aren't. I mean, websites are USEFUL - like, say, Google Maps or OpenStreeMap. But do I have fond memories of Google Maps? No, not really.
Fediverse, man... this is community. I like that. I like my Mastodon instance. I'll probably grow to like Lemmy too.
The irony of the web is that it opened up so much more rich expression than text did, so much so that nobody ever mentions the word "multimedia" anymore. And yet, because we are awash in rich expression, it all blends together to be sorta unremarkable, because that is the world we are in now.