Thought experiment (what would it do..?)
Movies are too often based on coincidences and plot contrivances, without with, the movie story just couldn't happen. So it is with this movie as well.
Let's imagine a slightly different set of circumstances. Let's say Sarah isn't in the city, but on holiday with her boyfriend, somewhere in Bahamas or Hawaii for two weeks or so.
The Terminator arrives at Sarah's apartment, but obviously can't find her, because she's not even in the whole city. What will it do? What are its options?
Does it do the nonsensical reckless plan of just somehow silently lockpicking two doors and throwing Matt around quite a lot, before killing him by the method of 'noisy property damage'*, and then proceed to search for female bipedal entities that are still around after all that noise? Not very 'infiltratory', not very intelligent, not very efficient, when you consider it'd be better to kill a secondary 'mob' silently, so you don't alarm your main targets.
The fact that the robot doesn't even figure out whether Ginger IS Sarah or not before proceeding to - SHOOT (why not throw HER around as well, why so inconsistent?! Why not shoot Matt as well? Think before you answer, because the explanation can't be 'because guns are noisy', considering how much noise the robot made anyway, throwing Matt through a friggin' WALL!) is just so stupid, but we'll put that aside.
OR.. does it try to just somehow implant itself in Sarah's apartment to wait for her return?
* (why would an infiltration unit that has intimate knowledge of the human body need to waste so much time and destroy so much, why would it create so much noise just to terminate someone, when we have been SHOWN him one-shotting and one-punching people dead before?
Furthermore, an efficient killing machine created by an A.I. intelligent enough to invent TIME TRAVEL would know about acupoints, chakras, oxygen for brains and all that. The Terminator could have so quickly choked Matt, pushed a couple of acupoints to kill him, in even less time it took to kill the punk rocker by punching his chest in and taking his heart out.. the Matt-fight makes no sense from almost any perspective you think about it, when you factor in all the 'musts', as in, this terminator robot MUST know fast and efficient ways of murdering human beings)