How can I see what visitors are searching for?
Unfortunately it's pretty limited.

View top search phrases (in Drupal 6.x):
yoursite.com/admin/reports/search

The data originates from the watchdog table, which means that how much data is provided depends on how far back your watchdog data goes (usually not far).

The Search Log module will provide more robust results. Or use Google Analytics, as the commenter above suggested.