The Pax Romana time machine can only be used once, in all of time, but it'll let us transport a large warehouse's volume of people and stuff to any point in humanity's past and make one divergent timeline. We're not limited by religious beliefs, or in fact anything but our consciences; we can do whatever we think will have the best chance to make a finer world.
We're going to need to keep it secret, of course. Trying to have a public vote or any kind of communal figure-it-out-together effort is almost certainly going to let whoever is wealthiest and most ruthless get there first and I think this is one thing actually every single person on Earth will agree is a bad idea. Except Vladimir Putin, who is in fact the wealthiest and most ruthless person and the one who will most definitely be the one who uses it and end up breaking the known universe like a piggy bank to grab the resources within for himself and his cronies.
But if we keep it quiet we can hand pick the people and the material that'll give us the best chance to accomplish whatever it is we're going to accomplish in our own time and make this a very stress free operation. So what do we want to do?
My thinking is going back about 60 000 years, to present-day Rwanda, or whenever and wherever it was that modern humanity got started. Figuring that out as precisely as possible is the first problem, and the people we recruit for that are probably going to be well suited for the rest.
The idea is to jump-start that group of 40-50 000 people even more than whatever strange thing it was that spurred them to spread out over the entire Earth, with books and solar-powered computers and malaria medicine and good shoes and whatnot, and convince them to put off the spreading out and stay together for long enough to a) develop the Internet and b) develop universal human rights.
My hope, as you may hope too, is that this will form the basis for a united human race that stays in touch as it peacefully travels the world and integrates with the neanderthal populations where they settle and uses natural resources responsibly and forms a stable, sustainable global democracy that would no more war with itself than you'd kick your brother in the nuts.
(Yes, it happens, but it's considered immature, inconsiderate and not a proportional or justified response to anything.)
Some problems that'll probably have to be solved in the planning stage:
We're going to have to hope our chrononauts and proto-humans can learn to effectively communicate with each other. They have bicameral brains and maybe more differences we don't know about, they may be physically prevented from having a language as we'd define it. The alternatives then are cross-breeding with a species on non-verbal humanoid which I don't have to tell you would be evil, or breed amongst ourselves to supplant them which, what are we even here for?
Then there's nothing we can do to stop them from settling over the world as they originally did and repeat the history of separatism and all that shit. We may be able to do something like Wakanda that'll put Africa in a position to defend herself when the colonizers come back, so in the worst case at least we'll avoid five centuries of chattel slavery and all that.
Now if things work out, I realize we run into basically the same problem when our uplifted cro-magnons meet neanderthals in the wider world. Now they have ethics and can't crossbreed happily as in the old timeline. I think we're going to have to have our interconnected global society side by side with these neolithic tribes. We can't even let them die out. We'll have to build reservations for them and look after them as if they were simple beasts.
Except they're not like other animals, no more than we are. This opens up some questions I'm sure we need those scholars to help us plan for before even deciding on this time travel thing. I don't mean just the ethics of forcibly keeping dangerous tools out of their capable yet innocent hands, but how we now have a sliding scale between "animals" and "humans". Ideally, I guess we're just going to have to go vegetarian, and extend the same rights and protection to the whole animal kingdom as to our hairier cousins and generally take more seriously our responsibility as stewards of this planet.
Now I'm sure I have offended racists deliberately by reminding them we all come from central Africa and friends of learning inadvertently by showing some glaring gaps in my knowledge in equal amounts, so it's time to shut up.