Continuity mistake: When Marty goes back from 1955 to 1885 and meets the Indians, note that none of the Indians shoots any arrows nor do any arrow hit the car. You can see that when he drives the car back into the cave. But when he gets back to the cave after the Cavalry ride, a long arrow sticks out on the side of the car.
Factual error: At the end of the train scene just before Marty travels to 1985, the train busts through a road block warning that there's only 1/4 mile of track left before the ravine. Keep in mind that the train is travelling at approximately 70-80 mph by this time. At 70 mph (and not even accelerating) it would only take the train 13 seconds to reach the ravine. The actual time in the movie is at least a minute.
Deliberate mistake: When Marty is being chased by the Indians, a wide shot with the DeLorean at the bottom of the screen coming towards us and the Indians following reveals the entire background for a mile or two. However, there are no cavalry visible anywhere despite the fact that they pass over the cave in the following shot only five seconds after the last Indian does.
Continuity mistake: When Doc's about to send Marty back to 1885 from 1955, and Marty's dressed up...when Doc's walking back and forth loading up the car - look at the shadow from the building. You see it changes by several feet in a few seconds as the scene was obviously shot throughout the day.
Factual error: In 1885, Hill Valley and its surrounding areas are shown as having a desert climate, when in both 1955 and 1985, the climate is grassland. In reality, Nevada County, California (where Hill Valley is supposed to be located) is and always has been grassland and forests, not arid desert.