Go to the Qatar Racing and Equestrian Club (end of October there is racing on Wed and Thurs). The Equestrian Federation is right next to it. You can also try to get a tour of Al Rayyan Farms
http://www.alrayyanfarm.com/indexFlashTrue_main.php
Send me a private message. My husband has some free time and has done some work at the race track if you'd like to visit some stables. :)
When are you going to be here?