Get something like this case for a laptop 3.5" drive: http://www.amazon.com/s/?ie=UTF8&keywords=hard+drive+disk+enclosure&tag=googhydr-20&index=aps&hvadid=24987842071&hvpos=1t1&hvexid=&hvnetw=g&hvrand=16830885933722929405&hvpone=&hvptwo=&hvqmt=b&hvdev=c&ref=pd_sl_26mr0v9mt8_b
from Best Buy, New Egg, or whatever your local computer store is. This enclosure will let you hook your old drive to your new computer.
Remove the old drive from your old laptop. Usually one or two screws from the bottom or maybe the side of the laptop. It's as easy as changing the battery.
Hook up your new external drive case to your new computer and insert the old drive into the case. Should be a USB connection.
Boot your new computer and you should see the second drive in your files.
Drag and drop the files you want to save to your new computer.
If you can't see your old drive, and I have never tried this with Windows 8, or Apple then do as BBadger suggests and boot from a USB drive into a Linux operating system. Knoppix, Ubuntu, Puppy, etc. Any one will work and it is easier than it sounds.
To do this, start your new computer while watching the screen. It should say something about hold F1 or F2 or whatever to get into your BIOS. Do that and set the USB drive as the first place the BIOS looks for an operating system.
You can buy a Linux operating system on bootable USB stick from https://www.osdisc.com/products/linux/puppy
and lots of other places. See Google. You could download the OS, but making a bootable drive is a problem for some. Good advice at http://www.pendrivelinux.com
Drag and drop files from your old drive to your new computer.
You don't need to, and probably don't want to, install the Linux operating system on your new computer but the option is there.
As a bonus you now have a second hard drive to back up files to and a new operating system to play with. Total cost is probably about $30.00.