The first url I actually remember would probably be liii.com:8000 as that was the host of a Merc MUD I played on(Barren Realms). I have some recollection of playing on it a few times before we got internet access at home. Now they're at http://www.barrenrealmsmud.com/ and I haven't visited in probably a decade now. Nice that they're still around.
The second early url I remember is crl.com, the dial-up ISP my dad signed up with. We had a shell account on a real Unix box, with a whopping 5mb storage, and we weren't allowed to run SLiRP/TIA so we couldn't emulate a SLIP/PPP connection and use graphical browsers(a sore spot as the Web got increasingly graphical). They moved to business-only services some years later (and we moved on to other ISPs) Now the domain appears to be taken by a drug company. I don't remember what crl stood for, even.
A third good one was ftp.wuarchive.wustl.edu. They had a pretty spiffy anonymous ftp in the early 90s. Apparently it is now "too expensive to push bits to the world."
My predominant memory of the internet before around 1996 was that the Web was only one of several viable systems: email, Usenet, telnet, and ftp were also used by my 8-to-10-year-old self. But as I got older, the Web rapidly took over everything. I was still using dial-up right up until 2003 when I went off to college. And then still used dial-up when home on breaks, until I got a laptop and stole the neighbor's wi-fi. It's funny to think of how much dial-up's user experience grew less tolerable over time.