VERY General:
Once your form posts, in this case, back to the same form, you will have access to all of the fields entered via the $POST array. For example, the field "airport" will be accessible as $POST['airport']. You need to build up your SQL query, using the values of those fields as part of the SQL. Using the airport as an example:
"SELECT * FROM flight_data WHERE airport = '" . $_POST['airport'] . "'"
Obviously, your WHERE statement will be a little longer, using all of your criteria (departure, destination, airport, time of day, etc).
In keeping with rachel's suggestion, I suggest you visit sitepoint. They have a very good, highly rated tutorial on building your own database-driven website, using PHP and MySQL. Strangely enough, the article is entitled Build your own Database Driven Website using PHP & MySQL. It's a long article (that page is the intro, and there are 4 parts, each 10 pages long), but it covers a LOT, including installation of PHP and MySQL. If you don't need that info, of course you can just skip to the next lesson.
Good Luck!