Continuity mistake: After Kyle Reese hot-wires the car and has the nightmare, he drives off and the car is fine. Later, when Sarah's date cancels on her and she heads to the theatre, Reese tails her and the hot-wired car has damage on the driver-side front. (00:20:29 - 00:24:58)
Other mistake: At the gun store, the Terminator asks for "The .45 long slide, with laser sighting." He then acquires the gun and indeed uses the laser sighting more than once. The Terminator should have no need for the laser sighting, since it basically has built-in laser sighting.
Revealing mistake: The T-800 can be seen wincing when Traxler shoots him. Apparently, the detonating squibs were painful for Arnold Schwarzenegger.
Plot hole: When Kyle Reese first enters the picture, the police immediately start chasing him (which ultimately works to the Terminator's advantage, as he is able to use the radio chatter to track his target). But why are the police after Kyle in the first place? All he did was steal a homeless guy's pants. There's no way that would generate that big of a response from the LAPD.
Suggested correction: Hardly a plot hole to have him be chased by the police. But they saw a half-naked guy in an alley at night; that's suspicious. He stole a man's pants and ran. Then he assaulted a police officer and took his gun. It escalated pretty quickly.
