Google Search and Own Website Search?

Why not create a form that submits direct to Google? This would be the simplest solution (assuming you want to search every website in the world):

<form action="http://www.google.com/search" method="get">
<input type="text" name="q" placeholder="Search Google"/>
<input type="submit" /></form>

I believe that would do the job you want it to. The result would look something like:

https://www.google.com/search?q=this%20that