You don't. You leave the first two be since they are both modern sci-fi cinematic classics. But this is Hollywood we are talking about, the people who will kick a dead horse even after it's just a skeleton.
So here is my idea for a Terminator sequel.
We have it take place in the future during the machine and human war. The idea of time travel has not been introduced yet, so this will take place during that exposition that Kyle Reese was giving to Sarah Connor during the first movie. At one point, he mentions that the machines were sending in terminators that looked like humans to be spies, similar to the Nazi's during the Battle of the Bulge. Here is the time where we get to explore some more of that time period and gives more weight to the desparation that both sides were feeling when they decided to screw with the rules of time and space by rigging the game.
So the story begins during the latter half of the machine war. John Connor has become leader of the human resistance and now it looks like the humans have Skynet on the ropes. Rather than start an all out assault though, the machines do what Kyle Reese was talking about, they try sending in machines that look like humans to spy on the Resistance. After a few times of trial and error, where the models have really fake looking skin, Skynet finally has perfected the art of creating terminators that look like humans. Not only that, but John Connor receives word that Skynet has also found a way to create perfect duplicates of already existing people, even getting them to perfectly mirror a person's personality, and their memories, similar to the robots from the movie The World's End. And much like The World's End, they dispose of the original host once the process is complete. The beauty of this is that, since their personality and memories will be copied into the robot as well, there won't be any sign of anything wrong with them or that they had been replaced by a replicate of themselves. Now keep in mind that this is not like the Marcus Terminator from Salvation where he still had a human heart and human consciousness, so he still had the ability to make his own choices, nor will it be like the new Terminator in Genysis which is a human that has been infected by a terminator or some crap like that. I haven't seen it but from the trailer, it looked pretty dumb. No, this is a terminator that wears the skin and personality of a previously existing person. Their purpose is to spy on the resistance, learn what they can, sow seeds of confusion among the ranks, and eventually kill the resistance leadership, including John Connor from within. This is also before the liquid metal technology was created so it can't take on the appearance of multiple people, it's limited to being one person. Take the historic Battle of the Bulge from World War II and add Terminators to it and viola, you have a Terminator story that doesn't get overly convoluted or continuously screw with the time line.
This was Skynet's initial plan for ending the war on their terms. John Connor also receives information that during their last encounter assaulting a Skynet facility, the machines captured one or more of John's inner circle to act as these ops. John doesn't know who has been captured, or how many, nor is he aware of how many may be posing as officers or foot soldiers. At this point in the game, Skynet is not aware of John Connor's importance, they just know he is the leader of the resistance, they don't know that he is the one that will spell doom for them. Nor do they know of the importance of Kyle Reese. In fact, Reese being involved in this story would be minimal. I know that making him a target would up the stakes, but it makes no sense for Skynet to view him as such a threat at this point because neither side has developed the capabilities of time travel. Maybe it can be hinted at in the end, but don't use it to set up future sequels because those of us who have seen the other movies already know where it goes from here.
On top of a level of paranoia spreading in John Connor's mind, he doesn't know how he is going to figure out who the 'tratior' is. he can't interrogate them nor can he just start executing people because unless he gets it right the first time, he's going to be seen as going crazy. And he can't make his concerns public because then Skynet will know he's onto them and they will force their hand.
And most importantly: DON'T SPOIL WHO THE TERMINATOR IS IN THE MARKETING!!! For such a big budget franchise, this seems to be the one mistake they are constantly making.
Now the argument could be made that this doesn't add a lot of suspense, since we know that certain characters have to survive in order to continue/start the story of the originals. And yeah, you have a point, but if hollywood clearly doesn't give a shit about continuously screwing up the backstory, then why should I be chastised for trying to keep it the same? And besides, the events of this story would add a level of desparation to what motivates our heroes and villains in the first two movies. Both know what is at stake when they send their players into the past now, and showing more of the machine war could shed light for the audience. Just don't try to screw up the timeline even more. They've tried it three times since T2: Judgement Day, and it just gets stupider.
And there you have it, a Terminator story that does not screw up the story line. It also adds a level of mystery and suspense to the story that was also present in the first two movies. Contrary to popular belief, the action scenes is not the only thing that makes Terminator good. It's also that there was an element of horror present in the first two movies when the machines first showed up in the present.
Also, here's a tip: when casting John Connor, let Michael Biehn play John Connor. Since he played Kyle Reese in the original, in kinda makes sense that John Connor would also look like Michael Biehn
It's almost like they could be father and son!
