I prefer All You Zombies also by Heinlein which I believe was also the basis of the movie Predestination.
The standard and simplest time travel paradox is going back in time 15 seconds and shooting yourself before you activate the time machine.
Nice thought experiment!
The paradoxes are easy enough to come up with; I'm still waiting for a story wherein the paradoxes are resolved in a believable way.
There was an intriguing (to my young mind) short story in an SF anthology I read as a teen - in the near future, they've invented a time machine, and the government raises big bucks by selling dinosaur-hunting trips to wealthy folks. The past has already been thoroughly investigated and the hunters are only allowed to shoot dinos that would have been killed imminently by predators, comets, falling into lava, etc. This would ensure that the hunt did not alter the past in any appreciable way.
As it happens, there's an important election the next day. The odds-on favourite for Terran leader is a fine fellow, facing the evil Klinkhammer or something like that. But the good candidate is a shoo-in anyway.
Back to the hunt: One of the wealthy hunters stumbles clumsily off the temporary pathway into the primordial mud. The safari leaders are furious, but conclude there's nothing for it and that no harm was done.
They return to the present, and everything is ... different. The world is grey and bland, the buildings are all monolithic, and the newspapers announce that Klinkhammer has won the election.
As it turns out, the clumsy hunter had stepped on a butterfly pupa.