The only one I remember was back in high school a bunch of us used to read this NSFW comedy webcomic called Sexy Losers. I think it's actually still going!
It seems like a high proportion of older web comics are still going, and I suspect it's because the market wasn't as saturated back then, so you could find an audience and possibly even make a career of it with relative ease (not that it was ever actually easy, just relative to the market now). So many of the ones I followed back in high school are still going.
Saaaame.... though a lot of the ones I used to follow are somewhat based on the artists' lives, so I could see keeping that going. Another one ended, but the artist keeps making new stories with different characters set in the same world, which seems more interesting.
I don't really remember the very first but Snailord's Snailogy, Yuu's Anti-Social Media and Evan Stanley's Ghosts of the Future are the ones that stand out most in memory when I try to think back.
GotF is probably the only one I know to still be sort of ongoing though the artist is busy working on official Sonic stuff. Fully understand the whole stuff you read inspiring you to create and it's Snailogy and Anti-Social Media (along with AKoW) that even inspired me to start my own comic here on Tapas.
I will say it's probably comics like Kikiine's Peony, celiere's Hidden in Plain Sight, Sinlaire's Only Human and other such story driven comics that really inspired me and are the reason why I want to be able to craft my own interesting and inspiring story based series. Plus like...a ton of others with pretty art that make me want to hone my craft more.
I don't remember what my first webcomic was, since it was a medium I went hogwild on for awhile on.
These are ones that I remember reading in 2004-2006:
Chugworth Academy by Dave Cheung and Jamal Joseph Jr. (Not sure if this one is entirely appropriate, skimming through it on the Internet Archive right now. Edit: Yeah, no. Aged really badly.)
Rachel the Great by Rachel Naboors (when it was on girls.com lol)
VG Cats by Scott Ramsoomair
Cyanide & Happiness by Rob DenBleyker, Kris Wilson, Dave McElfatrick and Matt Melvin.
Questionable Content by Jeph Jacques
Death and the Maiden by Nina Ruzicka
Charby the Vampirate by Amelius (gorey and maybe inappropriate, so fair warning)
Zap! by Chris Layfield and Pascalle Lepas
I also read The Devil's Panties Jennie Breeden, but I totally forgot about it until I saw it in this thread. And she's a mom now. 
I'm going to have to catch up on it.
In a way, a ton of these webcomics inspired me to start my own, but I got the final push from Lore Olmpus by Rachel Smythe.
The first webcomic I was introduced to was Zelda comic, somewhere around 2005~7. One of my friends in middle school discovered it and showed it to me, then we would read sometimes in the library during breaks:
All of my first webcomics were sprite comics that are all now lost to time. The first webcomic I followed that is still available on the internet? Probably VG Cats or Brawl in the Family maybe. It was all Nintendo-related stuff when I was a middle schooler though, because that was my sole interest in life for a bit.
Mine was Lackadaisy Cats! I was absolutely blown away when I found it on DeviantArt, something like a decade ago. It's still ongoing!
It did not inspire me to get into making comics, because it was so overwhelmingly beautiful and well-rendered I thought I would never be able to meet that standard! 
It wasn't until I came across modern webcomics with styles more similar to my own that I thought "Oh, this looks way more achievable!" and I began to entertain the idea of making my own.
My first webcomic isn't on here and I don't really have any images of it, but it was called the Dream Guardians. All the charcters were pretty much airlifted straight from a fanfic sooo
In general: it was "Stand Still Stay Silent". On tapas, it was "The Fingerbone of St. Cyneburg". "Stand Still Stay Silent" is still going, but "The Fingerbone of St. Cyneburg" hasn't been updated in a long while, so I think it's dead.
Think it was Off White back around 2007. On deviantart!
I wasn't huge into comics at the time but I admired the art, and thought it was pretty neat that somebody was making something like that (I wasn't even into wolves). Not long after, I came across Lackadaisy. That one was very nice, too. Still, I didn't show much interest in comics until maybe 2 or 3 years later. I was too busy with Harry Potter fanart, and in my spare time, World of Warcraft.
Also enjoyed Phoenix Requiem and others by that creator around that time. And I see mentioned, SSSS. Off White was definitely my first, though.
Probably around 2011 I became more interested in story telling and making comics of my own. Lacked good ideas and to be honest, satisfaction with my work, so it never went anywhere until recently.
