you need to relate the tables together and then use JOINS in your SQL statement to do what you want if I'm reading the question right. How to do this would be found in any SQL book, probably even an online tutorial. The ability to do this is what makes relational databases relational.
The trick here is that you need to join with your Airports table twice -- once for relating to the DepartFrom ID and once for relating to the ArriveAt ID. To do this, you must list the same table twice in the FROM clause of your SELECT statement, and assign the table two different "aliases" using the SQL keyword "AS"; one for each use of the table. You also assign aliases for the two Airport columns in the result set so you can tell the Origin apart from the Destination.
The exact query is as follows:
SELECT A1.Airport AS Origin, A2.Airport AS Destination
FROM Flights, Airports AS A1, Airports AS A2
WHERE DepartFrom=A1.ID and ArriveAt=A2.ID;
Advanced SQL tricks like this will be covered in Part 9 of my article series on SitePoint.com.
Give us a yell if this doesn't do what you want it to!