I'm going to copy one of my old comments here, so some of the things I'm calling recent here are no longer recent:

I'll start at the beginning. The first comic I was introduced to (by a friend in highschool) was Bob and George, a sprite comic using megaman sprites and characters. It was alright, but not spectacular. Next is Dominic Deegan, a webcomic about a seer. I recommend it, though it's over now. The author is of that is currently doing Star Power, which is something completely different but seems pretty good for now. Moving on from chronological order, here's a recommendation list, just the good stuff. I'm leaving out comics that were cancelled or appear to be dead before reaching a satisfying conclusion, though some of them are good too. In addition, there are some that appear to have disappeared off the internet and I have little memory of. Those will not be listed. In no particular order:

• Miracle of Science. Sci-Fi Drama. Concluded.
• Awkward Zombie. Non-sequential videogame humour.
• Cassiopeia Quinn. Humorous Sci-Fi Adventure.
• Cucumber Quest. Lighthearted Fantasy Adventure.
• Curtailed. Furry Newspaper-styled Humour.
• Ozy and Millie. Furry Slice-of-Life Wholesomeness.
• Freefall. Oft-Humourous Sci-Fi... Drama? The kind of Sci-Fi that really explores the meaning and consequences of certain kinds of tech (especially A.I.) from an interesting angle. Easily the oldest webcomic on this list, both in terms of its own age and how long I've been following it. It's also one of the longest-running webcomics on the 'net, so you've got a binge waiting for you.
• Out-of-Placers. This one's been going slow. Fantasy Drama of some kind that's really giving you time to explore this world it's been building.
• Damsels Don't Wear Glasses. Fantasy Something. Again, this one's been going slow.
• Dan and Mab's Furry Adventures (Primary Site, Katbox). A Furry Drama Adventure. Starts off genuinely bad. Try to read the first few pages to get introduced to the characters, but if you need to, skip to page 64 to get to the start of the first real story arc, which is also around when the art at least starts improving somewhat.
• Project Future. This is, for all intents and purposes, a collection of fanfics for DMFA (above). If you liked that, you might want to read this next.
• Daughter of the Lillies. Fantasy Adventures, still in its early chapters.
• El Goonish Shive. Humourous Transformation Drama. Modern-Day Fantasy.
• Eldritch. Werewolf Drama. Can't remember much about it, but I think it was good. Finished.
• Faulty Logic. Humourous furry webcomic about... something?I can't really remember. Concluded.
• Girl Genius. Dramatic Fantasy Adventure about mad scientists, in a Europe ruled by mad scientists.
• Go Get A Roomie!. Interpersonal Drama featuring queer folk.
• Grrl Power. An interesting take on Superheros.
• Gunnerkrigg Court. Dramatic Paranormal Highschool Adventure featuring gods and myths.
• Housepets!. Furry Drama/Slice-of-Live/Adventure (by turns). Also: Duh.
• How to Be A Werewolf. Modern Fantasy Drama. Featuring fluffy wolvos.
• KiLA iLO. Sci-Fi Drama. Aliens visiting earth searching for something hidden there. Concluded.
• Prequel. Fantasy (specifically TES) Adventure. Bends the format a lot, might be an acquired taste.
• Radioactive Panda. Concluded. I can't remember what it was about, but I can remember it left me wanting more from the author, which is the sole reason for this recommendation.
• Schlock Mercenary. Corny Sci-Fi Adventure. Another long-runner, has updated very close to every day for 18 years, prepare for a binge.
• Selkie. Merperson gets adopted. Drama ensues.
• Sue and Kathryn. Paranormal Humorous Adventure featuring a zombie and a wraith. By the same author as Selkie, it appears as a fictional kids' show in that universe.
• Skin Deep. Paranormal Drama/Adventure featuring a secret society of shapeshifting creatures.
• Slightly Damned. Furry Fantasy Adventure. Gets pretty dark.
• Snarlbear. Alice-in-Wonderland-style fantasy adventure. Again, gets pretty dark. Concluded.
• The Monster Under the Bed. Paranormal Romance(?) Drama. Relatively early stages.
• Strays Online. Dramatic Fantasy Werewolves. Concluded.
• TwoKinds. Furry Fantasy Adventure.
• Widdershins. Fantasy Adventure.
• Wilde Life. Sinister Paranormal Drama.
• xkcd. I'm not sure how to describe this one. It's something different every week, but it's geeky and good.