I'm going to take us through the timeline of the Terminator movies and try to explain what should happen as we go as best I can. I'm not going to do much research (probably just Wikipedia) for ages or years, so just bear with me. Spoilers below.
1) Sarah Connor is born. She grows up, meets a man, may or may not marry him, they have intercourse in which John Connor is conceived and the man dies or leaves. John is born, he grows up, Skynet is created at some point and become self-aware, and Skynet begins Judgement Day. John and some other humans survive and start a resistance to the machines with John as their leader. On the brink of defeat, the machines send the terminator back to 1984 to kill Sarah Conner.
1a) The terminator tracks her down and kills her. This causes John to never exist. He never becomes the leader of the resistance and Skynet never has reason to send the terminator back to kill Sarah. Therefore, the timeline enters a loop, creates a paradox, and the universe explodes. Luckily, this doesn't happen because:
1b) The resistance finds out about this and John sends Kyle Reese with a picture of Sarah back to protect her. He succeeds and the terminator is destroyed before it can kill Sarah. However, Kyle is killed as well and is unable to affect the timeline or the future. This restores the timeline, although Sarah now knows about the future and begins recording advice for John. Meanwhile the picture Kyle had of her is taken of her and given to her. This should be the normal timeline. However, 2 things affect it:
1c) Parts of the crushed terminator are found and CyberDyne is created earlier than normal. The only way this can avoid a paradoxical loop that destroys the universe is if Judgement Day occurs, John survives, becomes the leader of the resistance, and the terminator and Kyle are sent back to the exact same seconds in 1984 as they were in the original timeline. And
1d) During a night of passion, Sarah and Kyle have intercourse and John is conceived. This can lead to two new timelines:
1d1) John is now a different person, with different personality traits. This can cause him to be in a different location for Judgement Day and is killed in the explosions or he fails to become the leader of the resistance, thereby removing him as a target for Skynet to eliminate, so they don't send the terminator back in time to kill Sarah, which begins a paradoxical loop that destroys the universe.
1d2) John is now a different person, with different personality traits. However, Sarah trains him well, he survives Judgement Day and become the leader of the resistance. Skynet and future John send the terminator and Kyle back to the same exact seconds in 1984 as they did in timeline 1. Therefore this is the new timeline that continues the Terminator timeline for the sequels.
2) John Connor, son of Sarah and Kyle, survives Judgement Day, becomes leader of the resistance, and leads them to victory over the machines.
2a) Skynet sends the T-1000 back to 1995 to kill John. It succeeds. He never becomes the leader of the resistance and Skynet never has reason to send the terminator back to kill Sarah (or the T-1000 back to kill John). Therefore, the timeline enters a loop, creates a paradox, and the universe explodes. Luckily, this doesn't happen because:
2b) Future John sends the T-800 back to protect John. He succeeds, though both terminators are destroyed. Therefore the future continues as it should, however,
2c) John, Sarah, and the T-800 destroy CyberDyne and Miles Dyson dies. They also incinerate the terminator pieces from T1. This delays Judgement Day from 1997 to 2004. This can result in two scenarios:
2c1) These 7 years affect John, in which Sarah has died from leukemia, and he dies during Judgement Day or does not become the leader of the resistance. Therefore Skynet never has reason to send the terminator back to kill John, the timeline enters a loop, creates a paradox, and the universe explodes. This can only be avoided if:
2c2) Although 7 years later, John becomes the leader of the resistance and Skynet and future John send all terminators and Kyle back to the same exact seconds as they originally did. Therefore this is the new timeline that continues the Terminator timeline for the sequels.
3) John survives Judgement Day, becomes leader of the resistance, and leads them to victory over the machines.
3a) Since, in 2004, John has been living off the grid, Skynet cannot locate him, so it sends the T-X back to kill his future lieutenants.
3a1) It not only kills them, but locates John and kills him as well. He never becomes the leader of the resistance and Skynet never has reason to send the terminator back to kill Sarah (or the T-1000 or the T-X back to kill John). Therefore, the timeline enters a loop, creates a paradox, and the universe explodes.
3a2) It only succeeds in killing his lieutenants. This causes either:
3a2a) John to hire new lieutenants who are inadequate and they lose the war. Skynet never has reason to send the terminator back to kill Sarah (or the T-1000 or the T-X back to kill John). Therefore, the timeline enters a loop, creates a paradox, and the universe explodes.
3a2b) John to hire new lieutenants who are just as capable as the previous lieutenants, and they are on their way to defeating the machines, which causes Skynet to send the terminators back to kill John/Sarah/lieutenants. However, Skynet and John need to send all terminators and Kyle back to the exact seconds from previous timelines for this to work.
3b) Similar to timeline 3a2b, however the T-850 is sent back to protect Kate and John, though other lieutenants are killed by the T-X. (Therefore, look at timelines 3a2a and 3a2b to see what the death of these lieutenants means for the future.) It succeeds, however Judgement Day still occurs. Kate and John survive and live to become the resistance leaders. Skynet and John will need to send all terminators and Kyle back to the exact seconds from previous timelines for this to work.
4) John and Kate lead the resistance post-Judgement Day. Terminator Salvation does not have any time travel and takes place before the time-travel machine is used. However, John, Kate, and Kyle all have to survive (which is implied as the universe hasn't exploded from a paradox) in order for the previous movies to occur.
Now that I have explained about all four movies and my brain is sufficiently hurting from all of this 4th dimension thinking, I do want to state some flaws that are evident to me:
1) The fact that all terminators and Kyle are all sent back to the exact same moments from each possible timeline. (At least in Back to the Future the dates were inputted into the time circuits, so it is conceivable that it had some sort of memory that allowed Old Biff to travel to the same date the lightning struck the clock tower. BTW, though I've thought in depth about the BTTF trilogy, I can't really come up with any flaws in their time travel timelines, except for that Doc was able to make a steam-powered, time-traveling flying locomotive that worked better than the DeLorean.) So how did potentially 7 different iterations of Skynet from 3 potentially different years send the original terminator back to the exact same second in 1984 (and same locations) on its mission to kill Sarah? And even if you want to give the reason that Skynet has computer efficiency that allowed it to somehow accomplish this, how did John accomplish the same task with Kyle. What if Kyle arrived a day late and Sarah was already dead. What if Kyle arrived in the middle of a freeway and was struck and killed by a car, thereby allowing the terminator easy access to Sarah?
2) Everything above is subjective to what we know from the series. We don't know what effect the death of John's lieutenants and Kate's dad have on the future. We don't even know the effect of John's death would have on the future. It's possible if Sarah was killed (let's say by a car while crossing the street) that some Everyday Joe would rise to the occasion after Judgement Day and lead the human resistance to victory, thus making Skynet target him and his mother instead.
3) While Kyle and Sarah were having intercourse, Kyle's memories of John should have changed as, since John is now composed of Kyle's DNA, his personality and physicality should have been different.
4) Also, why doesn't Skynet learn? By T3, I would think it would have figured out: send the T-X through the portal, than destroy the machine before Kate can take possession of it and send the T-850 back to protect her and John.
I hope this clears up some of the confusion and I hope I did not offend anyone with all of my exploding universes (which I got from BTTF 2). I am also sure that I failed to think of some scenarios or other aspects of the series. I do believe this could have been all avoided if:
1) Kyle wasn't John's father,
2) The original terminator was crushed to dust and no parts were found, and
3) John and Kate's future lieutenants were not killed.
I also took a few of my ideas from the movie Looper, which I believe, though it was a good movie, had many time travel flaws and the universe should have exploded at the end.
If you have any comments on what I've written, please leave them. I would be happy to read them and even comment back to discuss the movies some more.
Thanks for reading this really long-winded explanation. If only I thought this much about important matters, such as school and work.