I like a range of webcomics, from a few that are silly and entertaining as individual comics, to others where the continuing story is the most important thing, although individual comics still have a punchline. And the continuing story can be dark or happy or frivolous or just entertaining, all in the same title. So you have comedy and tragedy and other types of drama, all in various combinations. Many creators create and post their comics on a regular schedule, which can vary from weekly to daily, with occasional breaks for vacations or conventions where they sell their printed books and other related merchandise. Tintin's adventures were originally published as regular installments in a magazine, and so were at least some of Charles Dickens' novels. This may help put webcomics in context.
I have sometimes posted links on Facebook to individual strips that are amusing in themselves, without requiring any context, but that might give people enough of a taste of a comic to motivate them to read through other pages of the same comic.
It generally makes sense to start reading at the beginning rather than the most recent strip. To find the first strip, look for a link to the comic's archive. Sometimes an icon of a double backward-pointing arrow will take you to the beginning. In many cases, you can see the artwork and plotting grow more sophisticated and complicated as the strip's creator gains experience. Don't give up on a comic if the initial strip falls flat. If you read a few more strips, you may find that you like it.
Writers of fiction are taught to write about what they know. Comic creators do the same, with many strips being autobiographical or semi-autobiographical, sharing the creator's embarrassing moments and romantic failures (and successes) with the audience. The main characters are often roughly the same age as the creators, although this isn't always true. Sometimes child characters are based on the creator as a child, and other characters can be based on a creator's parents or other family members. This may lead to interesting discussions at family gatherings, particularly if one of the characters based on a family member gets killed off, as sometimes happens.
I want to recommend a few, and add a note that if a comic is recommended by the creator of a different comic, it is likely to be worth reading.
About a rather silly group of college students who are housemates. They eventually graduate but are still pleasantly silly.
Drama and romance with a lot of humour and some sad moments. Unlike most comics, it has an ending which represents a resolution of sorts of some of the recurring themes. Despite the name, it's not all about wicca or tarot cards, although wicca plays a part in defining one of the principal characters.
The interactions of a group of 20-something people as they struggle with friendships and romance and as they become increasingly mature (but still a bit child-like at times). The website explains that the title was derived from requests the creator used to get at comic conventions to draw girls with guns (she sucked at drawing guns, so she drew them with slingshots).
The description is much the same as for Girls With Slingshots, but the two titles aren't copies of one another. Something Positive is a bit snarkier and a bit darker, but still highly enjoyable.
Octopus Pie is an ongoing comic series, started in 2007, about two women living in Brooklyn, NY – their jobs, social circles, and some problems.
Described as "the sci-fi adventures of a girl, a mad scientist, and a robot."
Also worth noting: Sometimes characters seem to be doing something self-destructive like drinking way too much. Remember, it's a comic, and the creator may be exaggerating for effect... or alternatively the creator is setting the character up for something bad to happen.
You can find many other webcomics through Wikipedia https://en.wikipedia.org/wiki/Category:Webcomics, lists like http://www.thewebcomiclist.com/ or hosting services like http://www.comicgenesis.com/ and http://www.keenspot.com/
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