Do the milkshake the milkshake do the shake
You can now connect your PS4 controller to your old-school PS3 without needing to use a USB cable, thanks to a recent firmware update. While the connection does lock out a few of the controller's newest features, it works well enough to play games, and that's all that matters.
You will need a USB charging cable as well as (obviously) a Dualshock 4 controller and PlayStation 3 console, turned off.
After you've paired the controller with your PS3, it will remain paired until you hook it back up to your PS4 (a similar process to this one). The controller will only control one console at a time, so don't worry about accidentally turning the wrong one on or off!
Unfortunately, at this stage, neither Sixaxis nor vibrate functions are compatible (the recent update has provided the ability to turn on your PS3 via the PS button, however). Even after going through the motions of getting it all hooked up, the console won't recognise the controller as an official Sony peripheral - but if you do want to use the new controller on the old machine, now you have a wireless option, too.