Though not among the first, the Sunday comics in the Herald were among the best. The most visually interesting of them was Winsor McCay's Little Nemo. Begun in 1905, this was the first strip with an episodic story line: the narrative continued from one weekly installment to the next. Zachary Chavez has digitized Little Nemo's adventures from the first on October 15, 1905, to the last on July 26, 1914. You can see them here:
Little Nemo in the Comic Strip Library.
Here's the first — October 15, 1905 — setting the scene for the others to follow. Little Nemo falls asleep and is called to journey to a place called Slumberland. Each comic shows his adventures as he tries to get there. Review the wikipedia entry on the strip for a full description of the plot line. Click the image to view full size. You can see McCay's potent imagination and his skill as an illustrator and you can see also the relatively high quality of the Herald's printing.
This is the third; it ran October 29. 1905.
This is the sixth — December 3, 1905.
December 3, 1905
The giant elephant of September 23, 1906, is particularly spectacular. This is the first of a three-week set. You can see the second and third here and here, respectively.
On July 26, 1908, McCay drew this walking bed.
Foxy Grandpa was the Herald's first Sunday strip. It appeared on January 7, 1900, a creation of cartoonist Carl E. Schultze drawing under the name of "Bunny."
Produced later the same year, Pore Lil' Mose was the first Sunday strip to feature an African American hero. Taking its humor from racial stereotypes, it's in keeping with attitudes of that time and not at all in keeping with our own. However, there's nothing vicious or much more than superficially demeaning in the comic. Here's a sample from February 1901. You can see that the artist's approach to drawing and visual design are less imaginative and more in keeping with what we consider normal comics than is McCay's.
The Herald's longest-lived comic strip was Buster Brown. Begun May 4, 1902, it too lacked the visual imagination and drawing skill that McCay brought to the Little Nemo strips, but it was funny and became very popular.
Both Pore Lil' Mose and Buster Brown were made by Richard F. Outcault whose Hogan's Alley was one of the first and best-loved Sunday strips. Outcault specialized in depicting antics and life-in-general among the urban under class of dwellers in tenement aparments. His first, in 1894, was a series these gritty cartoons for Truth Magazine.
In 1895 Outcault moved the Pulitzer's New York World and, while there, he made more prominent one of his Truth Magazine characters, the famous The Yellow Kid.  Here are a couple of these early Hogan Alley strips in which the kid appears.
This next comic shows the Yellow Kid in a Thanksgiving turkey raffle. It appeared in the New York Journal following Outcault's move there in 1896. What was Hogan's Alley has now become McFadden's Row of Flats and the presentation has moved closer to an illustrated story than a simple comic.
The Little Bears
The Yellow Kid
The Katzenjammer Kids
Winsor McCay. Extract: "Most of the prominent comic strips of the 1890s and 19-aughts are remembered today as pioneers. Little Nemo in Slumberland is among the first to be remembered for its outstanding quality. Even today, it is regarded as one of the high points in the history of comics. Winsor McCay was the son of Robert McKay (later changed to McCay) and Janet Murray McKay; Robert at various times worked as a teamster, a grocer, and a real estate agent. Winsor's exact place and year of birth are uncertain — he claimed to have been born in Spring Lake, Michigan in 1871, but his gravestone says 1869, and census reports state that he was born in Canada in 1867. He was originally named Zenas Winsor McKay, in honor of his father's employer, Zenas G. Winsor."
Finding "Little Nemo" by Douglas Wolk
R. F. Outcault
American comic strips before 1918
What's with the Strips, Anyway?
Comic Book Legends Revealed by Brian Cronin
Comic Book Resources
The Yellow Kid
R. F. Outcault, The Father of the American Sunday Comics
The Kid From Hogan's Alley By John Canemaker (Kitchen Sink Press, 1995)
Bill Bedard on “Little Nemo in Slumberland”
A hundred years later, fans are still over the moon about 'Little Nemo in Slumberland' comic strip October 22, 2005|By Kenneth Baker, Chronicle Art Critic
(Arguably) The Best Comic Strip Ever
Little Nemo in the Comic Strip Library
Little Nemo, a musical production; music by Victor Herbert, book by Harry Bache Smith (Cohan & Harris, 1908)
Comics Timeline, the history of the funnies in America
Chronology of Comic Strips and Comic Books in America
Comic Strip Library by Zachary Chavez
R.F. Outcault's "Pore Lil' Mose"
 Wikipedia defines the comic strip as "a sequence of drawings arranged in interrelated panels to display brief humor or form a narrative, often serialized, with text in balloons and captions," and says "the Little Bears (1893–96) was the first American comic with recurring characters, while the first color comic supplement was published by the Chicago Inter-Ocean sometime in the latter half of 1892. On January 31, 1912, Hearst introduced the nation's first full daily comic page in his New York Evening Journal." Here's the little guy:
 The first New York paper to feature a comic was Pulitzer's New York World (1895). The Herald's first strip was, apparently, Foxy Grandpa by Carl E. Schultze in 1900 (Chronology of Comic Strips and Comic Books in America). Here's a Foxy Grandpa strip from 1904. It does seem to display high quality color.
The Herald's claim on having the best comics comes partly from the superiority of its color work. Among cartoonists, the Herald's printing was said to be the best (The Kid From Hogan's Alley By John Canemaker (Kitchen Sink Press, 1995). It was also known for excellence in the quality of the comics themselves, as the Little Nemo strip shows, and for its innovativeness, as the Pore Lil' Mose strip shows.
 Unless otherwise noted, my source for the Little Images is Zachary Chavez's Comic Strip Library. Compare the first Little Nemo, in the Herald, with this strip, the Happy Hooligan, which also debuted in 1905 (in the New York American):
 See note 1.
Newspapers stole successful comics from each other. Outcault left the Herald for William Randolph Hearst's employ in 1906, and after a court battle, he continued his strip, now nameless, in Hearst papers, while the Herald continued their own version of Buster Brown with other artists. The latter lasted until 1911 or so, and the former until at least 1921.
 As the Yellow Kid grew more and more popular, William Hearst began trying to entice him over to his New York Journal. In 1896 he succeeded and Pulitzer struck back by hiring George Luks to draw his own version of The Yellow Kid. Both strips ended a year later. This was the era of yellow journalism which originally was referred to as "yellow kid journalism" after the comic.
 These Yellow Kid images come from The Yellow Kid on the Paper Stage by Mary Wood.