You would think the T-800 would stay on pursuit. They make the thing seem so human like in the movie. Machines would never stop to break into a car or go to a hotel. They would just keep pursuing with anything on them that could lead them to their target right?
There's a number of in-universe reasons for this behavior.
First, Terminators only had limited knowledge of the past, because Skynet had most of its historical data "destroyed in the war." For example, the only thing Skynet knew for certain about John Connor's mother is her name, not her age, address, or anything else to narrow down the list. This is why several Sarahs were killed before the right one was found. The Terminators did not know all of the immediate relationships a particular target had without acquiring this information somehow while in the past.
Second, the Terminators' artificial intelligence was highly advanced. A Terminator would break in to a car if it meant that it could increase its mobility during a pursuit. Terminators, while able to run long distances without tiring, were still mechanically not as fast as a car in a high-speed chase. A Terminator would choose to perform an action that increased the likelihood of mission success, even breaking in to a car or hotel. Human laws were irrelevant to the mission, and would be ignored.
Third, Terminators had a lot of data about human behaviors. They knew that one could get humans to cooperate by threatening people close to them, and that humans would seek out familiar places to hide if they were in trouble. Terminators had a lot of routines they could employ to extract information (e.g. torture). If a Terminator lost physical contact with a target, acquiring data to predict likely locations the target would go to is a perfectly logical step.
Fourth, Terminators did not have some of the fancy mumbo-jumbo sci-fi tech you see in other movies. They could not track a human that was 10 miles away, or place a "tracker" on a vehicle that had GPS and reporting capabilities from an unlimited distance, etc. While they could see and hear better than humans, these capabilities were still limited by the laws of physics. If they lost sight of the target, the Terminator would have to extrapolate the best way to find the target, which might include finding people the target knows, checking databases, etc.
Putting this all together, Terminators are not just mindless robots, but can actually determine an appropriate course of action for almost any given situation. Reconnaissance and autonomous action is just as much a part of their programming as simply "terminating," which is why these robots were feared by humans throughout the entire series. You couldn't easily hide or run away from a Terminator, you had to actually outsmart it--and it learned your tricks every time you used them, making it a terrible reality that either you or it would be destroyed/killed before it finally stopped.