You can download any VMware Fusion installer .dmg directly from VMware's web site from the link I previously provided however you do have to have a valid Serial Number for that version and a separate valid Serial Number for each copy of VMware Fusion you use on each computer if using it on more then one computer.
Since you are currently using Fusion v1.x the Serial Number for it will also work in VMware Fusion 2.x and if you're only going to run it on the new computer and remove it from the old computer then you do not need to purchase an addition Serial Number.
If the new computer has or you are going to install Mac OS X Lion then you will need to use VMware Fusion 3.1.3 (or higher when available) and also purchase a VMware Fusion 3 Serial Number or wait until the next major version VMware Fusion 4 is released.
As far as copying/moving files you can either use Migration Assistant or any other normal method to copy/move files between computers.