In The Terminator, Reese says Skynet had almost no information on Sarah Connor: Just her name, and the name of the city where she lived.
The T-800 arrives in 1984, finds a phone book, and dispatches the first two Sarah Connors listed. It then goes to Sarah's apartment and, in a case of mistaken identity, kills her roommate Ginger (and Ginger's brave but luckless boyfriend Matt). Then it hears Sarah leave a message on the answering machine and realizes that she is still alive. It locates Sarah in Tech Noir, and spends the rest of the film trying to kill her.
All well and good, however:
Did it know this was the right Sarah?
If so, how?
If not, why was it so determined to kill that particular Sarah Connor?
There were probably a few other women named Sarah Connor in the Los Angeles area who weren't listed by name in the phone book. In fact, it's sheer coincidence that the listing for Sarah's address was in her name and not Ginger's.
In trying to hunt down "our" Sarah at all costs, the T-800 puts itself at significant risk and suffers a lot of damage. Before assaulting the police station, would it not have made more sense to search Los Angeles for other potential targets?