For visitors that used my site's internal search function, where in Google Analytics can I see the Google searches (organic and paid) they landed on my site with?
In the Reporting tab of Google Analytics, head to Content → Site Search → Usage.
Then, from the Secondary dimension dropdown, open Traffic Sources and click Keyword.
Now your data will be organized first as either "Visits Without Site Search" or "Visits With Site Search", and then by the organic and paid search keywords that were used.
You can click the headings of the columns to order the data by visits, bounce rate, etc.
This can help you analyze what incoming search keywords commonly lead people to search on your site itself, and how they typically behave.
The same comparison can be made in the other Site Search sections, such as:
And other options in the Traffic Sources dropdown may be useful to group the data by, such as: