I'm making a simple website with a form where a user enters a word and then I take that word and google it using a php script and displaying the results. How would I be able to do this? or where would I get a script to do this?
You would look for Google Site Search - http://www.google.com/sitesearch/
I'm fairly sure reusing their results as you want to do is against their Terms of Service.
Why not simply create a form that redirects to the Google result page? Here is a tutorial on how.
First, as quick solution, you should redirect your users to Google's result page.
When your sript receives the search term, it may
Example: http://www.google.com/search?site=&q=site%3Astackoverflow.com+banana