Purchase several packs of note cards and write the city of the airport on one side of the card and the three-letter IATA code on the other side. The more effort put into studying the codes, the faster you can memorize them, so try to study the cards daily during down time or on a lunch break. Set a goal of learning a realistic number of codes in a certain time period and stick to it. A complete list of codes can be found online (see References).
Eliminate the ones you already know from the pile of cards so it becomes less intimidating. Many people will be surprised by how many airport codes they already know. A lot of the commonly known codes are named directly after the cities where these airports are located, such as MIA for Miami and ATL for Atlanta---the busiest airport in the world, with nearly 78 million travelers passing through each year.
Learn the most popular airport codes first or the ones that will most be used for your purposes. A list of the 30 busiest airports in the world can be found at the World Airport Codes website (see References) and provides a good starting point. While there are so many codes to know, be aware that flights from each airport only travel to a finite number of other cities, and it may be helpful to start by learning these.
