Oh man, webcomics. I definitely spend way too much time reading them. Props to @Lakarr for The Phoenix Requiem suggestion -- I loved that one. The following is a selection of the ones I read which would be appropriate to link here (i.e. family friendly/not R-rated, though if some of you other 18-and-older folks want suggestions in that vein, I have quite a few and you can feel free to PM me and we can talk comics!). I wish I could give nearly all of these a "highly recommended", but I have restrained myself and given it to just the best of the best.

Hark, A Vagrant - (PG 13 for mature themes, but reasonably family friendly)
Kate Beaton is one of the most hilarious comic creators to come out of Canada. She primarily does comics about or inspired by history and the strange and wonderful people from other eras. Great if you're nerdy and like making fun of history. HIGHLY RECOMMENDED.

Lackadaisy - (PG 13 for mature themes, occasional depictions of cartoon violence and drug use)
If you like the roaring '20s, cats, and sepia tone, this comic is for you! I am absolutely in love with this series because it combines many of my favourite things, and has some of the most beautiful art out there, not in full colour save for her occasional poster pieces outside of the main comic.

Family Man - (PG 13 for mature themes, sexual content, depictions of violence, drug content)
Another one for history buffs, this graphic novel is actually the PhD thesis project of the brilliant Dylan Meconis. It's a slow-paced, dark, and incredibly well researched comic. If you're looking for something a little lighter, her previous, now complete series is linked below.

Bite Me! - (As given by the author: appropriate reading for ages 10 and up! Warning: this story contains depictions of bloodshed, mild naughtiness, werewolf assault, and aggressive chickens.)
A "vampire French Revolution graphic novel farce" of the finest caliber, this silly and delightful story doesn't take long to get through but is really excellent.

Dresden Codak - (PG 13 for mature themes, depictions of violence and some allusions to drug use)
This is one of those comics that you need to follow either from the beginning (don't be put off by the less sophisticated early art, because it gets AMAZING later on) or from the start of a major story arc -- the author gives his recommendation on how to start reading it HERE. Hugely nerdy (can you see a theme developing in what I like?) and incredible world- and character-building. The ongoing storylines and one-shot/short runs are all great. Everything the man does is brilliant. HIGHLY RECOMMENDED.

Rice Boy; Order of Tales; Vattu - (PG 13 for mature themes, occasional depictions of cartoon violence and some allusions to drug use)
Evan Dahm is probably one of the most creative and brilliant comic creators working today. If you love epic fantasy tales like the works of Tolkien, this is a must read. I could read his work over and over again, and the art is beautiful. HIGHLY RECOMMENDED.

A Softer World - (PG 13 for mature themes, sexual content, descriptions of violence, drug content)
A really beautiful photo-essay style 3 panel comic which is alternately witty, observant, laugh-out-loud funny, absurd, wistful, etc. You have to read it to get it, and sometimes the ALT-text is the best part of the joke.

Bird and Moon - (I think this might be my first (only?) completely all-ages, family friendly recommendation...)
A cute science and nature comic which is also educational. I love using this when I'm teaching undergraduate biology classes.

xkcd - (PG 13 for mature themes)
Probably the most clever comic on the internet. Randall Munroe is a genius, and his comics on mathematics, science, life, and love are just perfect. You've probably seen his iconic stick figures before, but if you haven't read the entire archives, you're missing out. HIGHLY RECOMMENDED.

Abstruse Goose - (PG 13 for mature themes)
More nerdy goodness. Not quite as brilliant as xkcd (in fact, kind of rips off the style) but funny and insightful nonetheless.

Saturday Morning Breakfast Cereal - (PG 13 for mature themes, sexual content, depictions of cartoon violence, drug content)
This daily strip is a mixed bag, but is generally pretty funny. Some great nerdy comics, some real philosophical ones, some just stupid but funny ones, some that are kind of meh. Click the red button for a bonus panel or joke, which is sometimes the funniest part of the strip.

Dead Philosophers in Heaven - (PG 13 for mature themes, sexual content, descriptions of violence, drug content)
If you've ever studied philosophy or the history of Western thought, you will likely find this hilarious. If you haven't, you will either find it boring, confusing, or both.

Hyperbole and a Half - (PG 13 for mature themes)
A hilarious, rarely updated comic which just speaks to me. It can be really heart-wrenching, and her depiction of living with depression is SO TRUE. All the feels.

Supercakes - (PG13-ish? for mature themes and cartoon violence)
This one is pretty new, but I'm enjoying it so far. A comic about women with superpowers, with a diverse cast of characters displaying many different races and body types. It's pretty great, and LGBT friendly!

Whew! That's a pretty exhaustive list. If I think of any others (or start any of the ones I've been recommended), I'll probably put them in a separate post.