Terminator 1: Skynet dispatched a T-100 Terminator ( Arnold Schwarzenegger ) back in time to 1983, to kill Sarah Connor--mother of the yet to be born John Connor---who by fate, grows up to become THE para-military leader of human rebels, who win the yet to happen war between them and human killing machines. Simply put, kill the head of the snake before it bears a child.
Terminator 2: Skynet dispatches an upgraded Terminator hybrid, a T-1000 that not only specializes in instant production of sharp lethal stabbing sword like weapons, but is also adept in shape shift copying any living human it encounters. It's target: pre-teenage John Conner himself.
The Resistance, makes a counter move: Conner, an electronics whiz himself, manages to capture a T-100 ( Schwarzenegger )--and reprograms it to travel back in time and protect Conner---and his paramilitary mentor and mother, Sarah.
Terminator 3: If Skynet can't find a good "man" to do the job---send in a even more lethal woman Terminator to do the job right. Again: Skynet Resistance, led by Connor---sends in yet another T-100 ( three times the charm for Schwarzenegger ) to fight the killer machine aiming to kill an adult John Connor AND his future wife. It becomes evident that despite earlier efforts, Skynet DOES arise to destroy the world and aim it's bid for global domination by making humans extinct.
Answering the time travel loophole: Time travel technology does NOT allow non-living objects ( weapons or clothing, for that matter ) to travel: ONLY LIVING TISSUE LIFE can go back, which is why the Terminators are covered in genetically created living human tissue--and why they must look deceptively human, understandably.