Taking the computer to a computer shop is one solution. But you can try a few things yourself at home.
The most important part is that read operations don't affect anything, but write operations do. Think of it this way: a file on your hard disk is stored as a series of 0's and 1's. A hard disk contains a file table which maps a set of 0's and 1's to a file. When you delete a file, the allocated space in the file table is simply marked as deleted but the actual contents is not yet removed. If you perform a quick format, the same principle applies. If you perform write operations, the space from the entries marked as deleted can be used to write your new files to the disk and the old file becomes irrecoverable. If you performed a secure erase the data is also explicitly overwritten and file recovery is practically a no-go.
The easiest thing to do is to hook up the hard drive to another computer. Either by putting it into a USB closure or hooking it up internally. As long as the OS is on another drive you'll be fine.
On Windows, there's a freeware tool called Recuva which you can try. It is capable of recovering quite a lot of files. Simply scan the hard drive and wait for the files to appear. Then export them to another drive.
On Linux, the operation might become a bit more complex. There's an Ubuntu guide which contains a lot of useful information: https://help.ubuntu.com/community/DataRecovery