isn't that the gear icon on the top right of the character choice screen?
I think for the basic Bronze to Crystal tier, the rewards can be easily transferred over. However, the top 100 rewards are difficult to transfer over.
If the season is still continuing, then if the said player is at Crystal tier rank and at the top 100, those positions could still change because people are still battling for them. It's difficult to judge just because the person placed top 100 that they will stay in the top 100 when transferring out of that Datacenter. If the reward is guaranteed for the top 100 positions when transferring out, it means one placement in the previous Datacenter will be locked out entirely and cannot be contested for.
The rewards are currently only for the top 100. Do you think it's fair that the rank gets transferred, but if they are in Crystal Tier, their credit resets back to 0? That way, they would still have to climb back up. Likewise, for all the previous ranks, they will keep their rank (Gold III, Diamond II, etc.) but their stars will reset back to 0. I think that's the most fair result since they will still have to battle against other players if they want to climb up in that rank.
Edit: Alternatively, prevent transfer of rank in the last 14 days -- but drop their Crystal Credit by 500 to 1000 credits if there are more than 14 days left and the said player decides to transfer so they still have to battle other players in that Datacenter when climbing back up during the season or be booted out of their original ranking.
As others said, you can do that straight from your character select screen: click the icon next to your character name on the right, upload your character settings, and then you can download them on a separate machine by hitting the same icon and selecting download. Keep them updated and you won't have to worry about losing stuff. Official guide on the XIV lodestone.
The longer, more irritating way to do it isn't altering game files, just saving and replacing folders:
Documents -> My Games -> Final Fantasy XIV - A Realm Reborn -> FFXIV_CHR#
The CHR# folders are your data for your characters; the more you have the more guessing you'll need to do on which character it is, but if you copy those over and replace whatever folders are there when you install XIV on a different system, you'll have your original settings as well. Easier to use the method through the character select screen, or by using the launcher to make a FFXIVconfig.fea file to transfer over, but that method is also viable, just takes forever to figure out which character is which if you only want one character backup and not all of them.