There are two broad categories of Time Travel in fiction.
The one kind is problematic, and inconsistent. This is the kind found in the Back to the Future movies, the Terminator movies, and most Star Trek TV episodes. In this kind of time travel, you can go back in time and alter history. There is some nuance as to whether the person doing the changing remembers the 'original' history, or remembers the new one. Of course, this kind of time travel gives you all sorts of paradoxes, like killing your grandmother before you were born (I use this link because it's there.. more accurately it should be "killing your grandmother before your father was born").
The other, consistent form of time travel is found in Bill and Ted's Excellent Adventure. In this kind of time travel, history already incorporates the changes that will have been made by time travellers. So, Ted 'Theodore' Logan, says 'Remember the trash can', the trash can indeed appears above the villain's head. Similarly, Ted's dad's keys are missing from the beginning of the movie, because in the middle of the movie, he goes back in time to get the keys and leaves them behind the police station sign, where it's been all along.
Although widely regarded as a silly movie, Bill and Ted's Excellent Adventure is one of the few movies with a self-consistent logic in Time Travel. Another is 12 Monkeys, and perhaps Star Trek IV. In books, the subtle form of time travel that happens to Arthur Dent and Ford Prefect when they get stranded in prehistoric Earth in the Hitchhiker Trilogy (by the late, great Douglas Adams) is also of this second type.