Can I see what a person who typed in a specific search word did on my site?

Yes. Login to your account, and go to Visitors: Visitors by search term.
Here you will see all the recent visitors to our site by search term: both the search engine, and the search phrase typed in.
From there it is easy to see what they did on our site, by clicking on the green or red puppets; this will take you directly to their click-stream(s).