I dreaded this topic. I think I've seen one, maybe two cases wherein time travel was used as a device in a story and there *wasn't* some kind of paradox created thereby. As a result, I've lost my taste for most time travel in movies and books. Which wasn't actually the question, I know. The question for the Word Whores was if time travel were possible, when and where would we go?
Easy answer. I'd go to all the points in past history over which there is some enduring mystery. When Plato was yammering about Atlantis, I'd be there to say, "Dude. Coordinates?" When the Nazca lines were being created, I'd want to be there asking, "Why?" I'd want to go back far enough to see whether the Sphinx really was sitting on the Giza Plateau ten thousand years ago when (and if) the area had a wet, temperate climate. So many things I want to know - not to 'the best of our knowledge'. That's not knowing. I want to know for certain what those inexplicable passages in the pyramids are about (without - you know - having been entombed there, myself). What else? Count me in for having a peak at the dinosaurs. And if I could arrange to safely watch that massive inland glacial damn break and drain most of North America during the ice age? I'm curious enough in general to get myself into some trouble transporting far enough back that the atmosphere of the planet might not be breathable. But heck. I'm scuba certified. I could get around that, right?
As for the future, sure I'd go if the chance offered itself to me. But I'll let you in on a secret. I don't want to know the future. Once you know the future, it's immutable. You can't change it. If you *had* changed it, what the future considered history would have changed, too, right? Suppose you go into the future and find out you died sad, broken and impoverished. Not knowing that, you could fight it with every fiber of your being, but if you've traveled to the future, found out that history says, "So sad. Died too young, in dire financial straits, never could do anything right, never amounted to much." How depressed would you be? You're hearing not what might be - you're hearing what happened. Done deal. What's left to strive for? After all, you can't succeed. The reverse is true, too. You show up in the future and you're famous (or infamous, depending on your preference). You cured cancer. You won the Nobel Peace Prize. You find statues of yourself in all the parks. When you got home, how hard would you try to live up to that? Would some of the joy be drained from your work and your life knowing that you can't fail?
So. Do I want to know when and how humans meet up with life from other planets? Heck, yes. But boy, I really don't want to know that three generations from now another asteroid clobbers planet Earth and sends humans the way of the dinosaurs. And I certainly don't want a Planet of the Apes reinactment. Some things, I'm better off not knowing.