if its steam, then you can backup the games on the old and then use those files to restore the game to your new machine.
I'm not seeing a similar mechanism in origin.
If they are physically installed then you need to reinstall them
save gaves can be virtually anywhere, often in 'my docs' or 'my docs/my games'
is your new machine a laptop or desktop, if a deskop then move the old hdd to new machine, if a laptop, buy a decent sized usb stick and tranfer that way.