The best way to set this up depends largely on what kind of camera you have. So I am going to cover a variety of methods that you can use with different cameras.
Step 1: Connecting to a Camera with a Remote Shutter Switch Terminal
If you are fortunate enough to have a camera that has a built-in shutter switch terminal, then it is really easy to connect an external circuit to your camera and activate it remotely. All you need is a connector that matches the terminal on your camera. Different camera models use different connectors. So here is a link to a good page that shows the various connector types:
For example, a Canon EOS Rebel T3i uses a 2.5mm stereo plug (3/32" can also work). As shown in the picture, the sleeve is connected to ground. The center ring activates the focus function. The tip activates the shutter. To activate the focus and the shuttle, you just need to connect that pin to ground (the sleeve). You can do this with either a mechanical button or an electrical circuit.