This thread has the list of shops :
http://rcqatar.com/rc-in-qatar/hobby-shops-in-qatar/It is good to start with electric cars. All shops have them. Do ask specifically for used ones. The ones running on nitro fuel based engines are more common though, but they require some knowledge to use and maintain them. Also, they make a lot of noise (which is part of the allure of them, as long as you run them in a proper location!)
A 4 channel indoor co-axial helicopter is another option. Models like the LAMA v4 are good choices.