While I was aware of the existence of modems and computers connecting to other computers pretty early in life, and I also knew of the BBS systems which people could dial into with their computers, it wasn't really until the early 90s that I heard of the idea of the "internet." Or, as it was called "the information superhighway." Al Gore had been touting it, which is probably where a lot of Americans first heard about it.
Most people didn't have access to the internet at first. Personal computers were growing in popularity, but without modems or any real set up to connect to the internet. My dad had Compuserve, which had some features, but we had to add the Mosaic browser before we could log on to websites on the World Wide Web. But his modem was kind of slow, like 9000 baud or something, as I recall. Later on, he got a 28,800 modem, which was a step up. I myself got a 686 which had only a 19k modem, and I signed on to a local ISP, but then went on to AOL. Then I switched over to Earthlink and got an IBM with a 56k modem, although DSL and cable modems were also coming on the scene about that time. In the early 2000s, I switched over to a cable modem and have been on cable ever since.
One thing I remember from the mid to late 90s was going on to Usenet, which wasn't the WWW but various message boards (called "newsgroups"). It used to be part of the basic internet package offered by ISPs, although at some point, they started phasing out Usenet or making it an extra add on. A lot of the groups were unmoderated and could get rather wild. I believe Usenet still exists, although I haven't really bothered to check lately.
When the internet first came on the scene, I don't think many people knew what to make of it. Some people were scared to get any kind of internet connection at all. I knew some people who refused to get into it at all.