The very same thing happened to me about a month ago. Once I got my new computer set up, I moved all my files from the old hard drive (which should be perfectly fine) onto my new computer and reformatted it to use as an external backup. As has been said above, you can get a USB connector very easily.
As far as removing the hard drive goes, that's very simple. I'm not sure how it works on the new Macbooks, but if you have an old (white, removable battery) MacBook, the process is super simple. All you need to do is: 1.) remove the battery, 2.) unscrew the 3 screws you see nearer to the middle of the computer with a mini Phillips head (removing them and the bar they're attached to exposes the RAM cards), 3.) pull out the hard drive using the small tab and, 4.) remove the drive from the mounting bracket with a mini hex screwdriver. You can then attach it to the USB connector and move your files.