Yes use a usb stick. Note that for some content you can only move and not copy (DRM).
Just move to the USB stick then move again to the new machines HDD.
If you don't have the firmware to allow the use of the USB stick on the old console you can download the firmware to it, just connect the console up to your router. You don't need XBL Gold to get the system updates
Recently done the same.
We found a usb-to-usb cable for £40 in currys or a 500GB hard drive for £39.99, also in currys.
It works fine, however the xbox will only allow the hard drive to hold 16GB of data when formatted... when you plug it into a comp you still have access to the full 500GB after wards though
Some data you can't 'Copy' but it lets you 'Move' the data instead - certain game saves etc