I'd guess just JavaScript maybe with JQuery (and AJAX) for the animations. Take in a string via GET, use a timer to animate moving the "mouse" (which is just a picture, not the actual mouse), and use a timer to "type" in the string one character at a time, then move the mouse some more and "click" the submit button (which isn't a real submit button, the whole thing is a form that doesn't do anything). Finally, it redirects the page based on the string from the very beginning.
You don't really need any particular program to do it, notepad could do it. I use
NetBeans for JavaScript (and most other languages except Perl and Android/Java) just because it'll show me any mistakes in syntax.
Notepad++ will also work for JavaScript and it'll be simpler for someone with less experience programming.
Still, I don't think that's the sort of site you'd want to make for your first script. I learned JavaScript from
W3Schools. They've got lots of examples you could go through until you get an idea of how to implement it into something you want. You want a firm understanding of HTML before you start JavaScript, then a firm understanding of JavaScript before you start AJAX or JQuery, so it's probably not something you can do in a day or two.