Seems like a lot of people want to know about time travel - which, given the objectives here, isn't all that surprising. I'm not a mathematician or a physicist - if you want the grounding in temporal mechanics, you're better off asking someone else. I
am a time-traveller, though, long before I ever came here. I've timeslid enough to have lost count of the number of times. So while I can't tell you the science, I can tell you the practicalities.
People who don't deal with this sort of thing generally have one of two concepts of how time works. One is that time is perfectly linear and immutable. If you go back and time and change something, you will have
always gone back in time in changed it, and therefore nothing has actually changed. It will have always already happened. The other view is that every choice splits the universe, into an infinite multitude of realities in which every choice has happened at least once.
The reality, from what I've learned of the mechanics, and what I've experienced, is somewhere inbetween. What the Initiative seeks to do - it's entirely possible, given the right skills. There are nexus points in time, moments that are important whether or not they seem so at the time. History hinges on those moments, and so does the timestream. If a moment is uncertain enough, two different futures will exist for the same timeline simultaneously - call it timeline 1, futures a and b. 1a and 1b will exist in phase until the nexus point snaps into certainty, at which point one will become the future of timeline 1, and the other will become a different reality, timeline 2. Or something else happens entirely, if things are uncertain enough, and both futures break off into timelines 2 and 3, and something else becomes the future of timeline 1.
I don't know anyone who's actually tried to carry out the grandfather paradox, though I assume if you did, you'd strand yourself in a timeline that was no longer your 'native' one - your reality would continue on without you while you're stuck in the other one.
And, I suppose, the one piece of advice I can give - if all of this seems overwhelming, it's simplest to think of things in terms of your own personal timeline, the chronology you've experienced. Which may be out of order with what people around you experience, but it's easier to keep straight that way. And even if you experienced things in a timeline that later - from your point of view - was broken off and orphaned, that doesn't make those experiences any less real.
[Aaaand now you are free to ask questions, even if he forgot to actually invite them. He's not really a teacher, okay]