I definitely tried my hand at a mud long before there was online anything else, but it was good ol telnet and just way too archaic for me. To this day, my GF stills plays them, which is just crazy to me.
The first serious online gaming I remember playing was the Xband on SNES and Genesis in 1994 ish. This was before even the internet really took off, but technically it was still online. For those who havent heard of it, it was this really spiffy modem cartridge that you inserted into the SNES, and then inserted the cartridge on top of it. Its been so long, but I remember is being SO badass. Most games were 100% lag free, although street fighter 2 had a little bit. Yet it was still so, so awesome for the time. Matchmaking was quick and automatic, and it kept stats and everything. Your phone bill definitely hurt though. It had some funny quirks - matches would come in as actual phone calls directly from player to player, so you had to tell everyone else in the house not to pick up the phone if it rang. So many times youd gbe the caller, hear it ring on the other end, and then someones mom picks up, and you hear the kid yelling at her.

Better yet, if you had caller ID, and someone disconnected on you midgame so they didnt get a loss, you could call them up and chew them out. Every now and then me and my friends would thrash someone, call them up and just start talking sh*t to whoever picked up....ahhh, good times. It was SO ahead of its time - Xbox live owes everything to it IMO.
As far as PC over the internet goes, I remember trying to play duke nukem 3d over the MPlayer service, but it was such a terrible mess, it took forever to find a game, took 50 years to connect, and then you had a 300-400ms ping in a game that was never meant to be played over the net in the first place. It was so horrible that I dont even want to count it.
It was Quake 1/Quakeworld that first did it right on the PC. I remember reading in some mag to download this nifty, tiny, non-bloated little program called "Gamespy".

It would just search for servers, I'd pick one, and just log in and play. My first thought was "This cant be this easy, and free, too?". If its too good to be true, it probably is, and after being so sour from Mplayer I almost didnt bother trying. It did indeed work as advertised, and was awesome.
Then the first real 3d accelerator came out - the Voodoo 3D. Online + 3D accelerator basically made me shun the entire 32bit generation of consoles, it rocked so much.