Start simple and get a handle on the kind of code you will need. First have a look at some tutorials. They will demonstrate how to wire things up and program for it. I think this is a good starting point right here.
Once you get the hang of that you'll need to loop into the best ways to connect your servo into the arduino so it can be operated safely without damaging the arduino. Then look into how to program it and see what you can come up with to combine the servo code with the button state change code.