If you book on AA.com, you have one (1) and only one opportunity to see your fare basis code. When you get to the "itinerary summary" page, you will see a button that says "Detailed Fare Rules." Click on it to see your fare basis code(s). I always print out the complete rules in case I need to refer to them when making a change.

If you use a travel agent or need your fare basis code after you book on AA.com, you can do the following:
1) call AA with your PNR at the ready
2) go to any AA ticketing location, give the agent your PNR, and ask for an "itinerary receipt." The printout will have your fare basis code(s), as well as the new codes for any mileage upgrades (e.g. "NXWWEB14/ALUG54"). Also it will show your fare calcuation summary.