Shymkent eats well. The city's large Uzbek population means the plov here is widely held to be the best in Kazakhstan, and the bazaars and roadside ashkana (cheap canteens) are the place to find it. A plate of plov in a working bazaar canteen at lunchtime costs a fraction of what it would in Almaty, and the rice is usually better.
The other regional specialities to look for are laghman (hand-pulled noodles in lamb broth), manty (large steamed dumplings), shashlik (skewered grilled lamb), and horse meat dishes like kazy sausage and beshbarmak (boiled meat over flat noodles). The latter is closer to a ceremonial dish than an everyday one but most of the larger Kazakh restaurants will serve it on request.