Shymkent is a city in southern Kazakhstan.
Ask your driver to drop you off at Mega Center, which is a hub for public transport.
South towards Tashkent (UZ)
Take a bus to the Zhybek-Zhol (Жыбек-Жолы) car market located in the southern part of city, e.g. buses 35 or 111 the stop "Ogonek" near the Shymkent Hotel. From the Zhybek-Zhol bus stop walk a bit further along the road and then hitch from there.
Places to Avoid
Avoid the Old Town (Старый город) at night, which has a high crime rate.