I got access to email through the SöderKOM BBS in 1994. In early 1995 I got a dial-up connection to the Internet via Algonet, an ISP started by my childhood buddy Ragnar. And in June of that year I put up a web site. It was a hand-coded static HTML page. A clearly recognisable version of it is still on-line after 20 years! But I haven't updated it since 2009.
My site was one of the first to mention archaeology in Swedish, so for many years it had an absurdly high search-engine rank despite its rather modest contents, beating out the National Heritage Board and most of the country's universities.
I started at the same time around 1994-1995.For me it was email that was the great news, making it possible to send mails all over the world instantly. The problem, not many had email accounts around these years. No firewalls and no antivirus and terrible surfing speed. Those where the Days.
My only claim to Internet fame is that I played one of the very very first MUDs when I was at Lampeter. In fact in my third year (1988/89), I spent more time in VAXMUD than I did in Wales, including indulging in a spot of cybersex, IIRC.
The closest I have come to internet fame is personally knowing (through classes and living in the same dormitory) one of the people mentioned by name in The Cuckoo's Egg, specifically the epilogue where the narrator was one of the people fighting against the Morris worm in 1991. (No, I have never met Stoll himself.)
E-mail was widely available at US research universities by the mid 1980s. But access was difficult for ordinary people before the 1994-95 time frame, and was still quite expensive at first. I didn't have a home ISP until 2004; it made no sense for me to have one because I had university access, and I could dial in if I was feeling too lazy to go to the office, where I had a T1 line available at a time when that was a big deal. The main thing that prompted me to get the home service was that DSL became available in my area. I still have that DSL line, because the other option locally is through a cable provided by Comcast, a company with a reputation for being evil even when compared to other US telecom companies. (FIOS is supposedly available in parts of Nashua, the state's second largest city, which happens to be located along the Massachusetts border, and nowhere else in New Hampshire. It's definitely not an option in my town.) That and I don't own a TV, so the cable/internet bundle never made sense for me.
'94, when we browsed newsgroups on the vt100 terminals while the CS guys proudly showed off their new "Mosaic" on the Sun workstations. Lynx with pictures!