Time travel stories have existed for millennia. Such diverse sources as Hindu mythology and the Talmud include proto-time travel narratives. However, these ancient tales lack the sophisticated analysis or even acknowledgement of the philosophical and physical questions evinced by the application of time travel. Our modern conception of the time traveler as an instrument capable of changing history arises in the birth of modern English science-fiction in the 19th century.
Perhaps the first widely famous fictional time traveler in the English-speaking world was Ebeneezer Scrooge from Charles Dicken’s A Christmas Carol, who travels to past and future Christmases as an observer. Time travel was not the main thrust of Dicken’s work, and the mechanics of time travel are wholly irrelevant to the story. Further, the fact that Scrooge is merely an observer precludes the work from tackling the thornier issues and paradoxes of time travel. Though well-received and popular,* A Christmas Caro*l did not excite popular interest in time travel so much as it excited interest in Christmas and the terrors of poverty in industrial societies. Time travel as a tool and theme in literature languished for decades.
Some forty years later, Edward Everett Hale introduced the English-speaking world to a slightly more active time traveler in his story, “Hands Off,” published first in Harper’s New Monthly Magazine in 1881. This parable features a time traveler who dares to meddle in the past by killing a stray dog in an effort to save a man’s life – only to find that his simple action has vast unintended consequences, eventually resulting in the destruction of all mankind. Hale, in one of the first works of its kind, expounds on the same theme that writers have been inculcating into the minds of science fiction readers for centuries: “Not a rain drop in that shower yonder but balances against a dust grain on the other side of creation” (Hale, 37). This is the progenitor of the butterfly effect: from the death of a single, seemingly inconsequential animal, the entire course of history is altered. Hale’s impetus for the story was religious – he hoped his work would demonstrate that, “From what I call evil [god] educes good” (Hale, 39), rebutting his contemporaries’ arguments against the benevolence of the deity. Perhaps it was because of this religious and overly simple interpretation of predestination that, “the story passed without comment in its day” (Science-Fiction: The Early Years, Everett F. Bleiler, 325). So too does Hale undermine the power of his story, by having the incident occur on a “not-really-real Earth,” as Nahin points out (Time Machines, Paul Nahin, 54). Nonetheless, Nahin, at least, argues that “Hands Off” was influential in the development of the burgeoning subgenre.
True public attention to the concept of time travel was achieved shortly thereafter, when Twain published A Connecticut Yankee in King Arthur’s Court in 1889, and a large readership got a full taste of an active time traveler who both interacted with historical personages and attempted to change the course of history. H.G. Wells story “The Chronic Argonauts” and his far more influential and widely read novel The Time Machine were published shortly before and shortly after Twain’s work respectively, and presented the public with time travelers that interacted with the future residents of the Earth.
While more can always be written, you may be interested in the answers provided in the FAQ addressing this question. Shout outs to u/sunagainstgold and /u/keyilan, both of whom provided answers there in the FAQ.
Another user, u/seasrocksandtrees, provided this post not too long ago which links back to even more answers here in this sub, providing additional information about early depictions of time travel in history.
The earliest instance of "scientific" (as opposed to magical) time travel I've been able to track down (and believe me, I've done quite a lot of tracking) is the short story *The Clock that Went Backwards" by Edward Page Mitchell, published 1881.
Mitchell's work is full of early examples of now-well-known science fiction tropes. The Man without a Body (1877) features one of the earliest technological teleporters (called a "telepomp", since Charles Forte wouldn't coin the word "teleportation" until 1931), as well as one of the earliest Star Trek-style "transporter accidents". The Crystal Man (1881) features a man turned invisible through scientific means, sixteen years before H.G. Wells. The Ablest Man in the World has one of the earliest fictional superhuman cyborgs and what is probably the oldest fictional artificially intelligent computer. The Senator's Daughter (1879) takes place in a futuristic 1930s which despite its Victorian trappings has a very proto-cyberpunk vibe to it and contains some of the earliest references to cryogenic pods and food pills, and also features near instantaneous ultra-high-speed subways linking every major city in the continental United States, concerts live-streamed to wireless speakers, fax machines, a 24-hour news cycle, and (most fantastic of all~) electric indoor heating.