Most stories about time travel are exactly that, stories. They're meant to be interesting to a reader/viewer/listener, not realistic depictions of what time travel would be like. They have existed for thousands of years; the most prominent examples are Greek mythological depictions of prophets (not so much time travel as the transmission of information through time), but there are many others. I think they represent a normal human reaction to regret and/or wish fulfillment, and this natural outgrowth from the human condition is why they become fodder for storytellers. I should also state my opinion that, especially in the realm of science fiction and fantasy, time travel is the laziest and most hackneyed form of storytelling, and should be approached with caution. All too often it becomes a convenient way for characters to be put under extreme stress or have a moving death scene and yet still turn up in next week's episode.
If time travel were to be real we would most likely see one of two general types. The first is time skipping forward, basically like suspended animation. Orson Scott Card did some interesting writing about this in the Worthing Chronicles. Basically, suspended animation technology led to a culture in which the wealthy would live for centuries or millennia by only experiencing one year out of every however-many-they-could afford. I think this vision is pretty accurate to how some people would behave, although the story tended to be a little myopic in its focus on just that element of the story and a handful of others.
The second is not so much time travel but the ability to cross into alternate dimensions, and in this way cross to a dimension that is chronologically out of sync with our own. Although crossing into alternate dimensions is itself way too science-fantasy, this method would bypass all the nonsense about causality and fate and whatnot, which to be honest is just a different form of wish fulfillment, that of a desire for personal belonging and purpose in the universe. The big stumbling block that is often ignored in stories of this type is what happens when you encounter yourself, and bad writers typically write the other selves out of the story, magically, which is a total cop-out. Of course it also raises the question of why limit yourself to just traveling to an out of sync time if you have all alternate realities to visit? Why not pick one where all your wishes have been fulfilled and go there, or for that matter pick a different one very week? This is basically the Bioshock Infinite route.
Good media about time travel:
http://www.escapistmagazine.com/videos/view/loadingreadyrun/1744-A-Stitch-In-Time
Bioshock Infinite (game)
Primer (movie)
The Back to the Future trilogy (movies, cute and non-serious)
Steins;Gate (anime, cute and non-serious)
The Worthing Chronicles / the Worthing Saga by Orson Scott Card
And of course various episodes of Star Trek the Next Generation, Futurama, the Simpsons, Family Guy, etc.
