How much autonomy could slaves attain and what did slave owners do to control them ?
Slaves had no real autonomy. 'Autonomy' is the right of self-governance and self-determination, which can never really be granted to someone who is regarded as property.
That said, some slaves did have more freedoms than others. This was especially true in states, such as Kentucky and Virginia (before Nat Turner's 1831 rebellion) where conditions for slaves were less harsh than in the Deep South. Turner provides a good example of the relative freedom that some slaves had. Turner was unique, due to his reputation as a slave preacher. Therefore, he was allowed to go to other plantations to proselytize. He was also allowed to take walks off of the plantation by himself. Some slaves were allowed to do this, though doing so required a written pass from one's owner to present to slave patrollers.
Slaves were sometimes also allowed small plots of land to garden. Some were taught to read and given permission to read, despite the fact that this was illegal and largely frowned upon by neighboring slave owners. They feared the dissemination of information to their own slaves.
Free blacks in the South were granted their freedom by their former owners. The slave system was one in which whites entirely determined the parameters of black people's lives -- how much "freedom" they could have as slaves and whether or not they could be truly free at all. There is no real autonomy in such a system.
Owners controlled slaves in many different ways, including psychologically and physically, which prevented them from gaining autonomy. Many slaves never learned how to read or write, as some southern states forbade them to. This limited their autonomy - the act of having rights and the ability to take control of one's own freedom. Since slave owners considered their slaves to be their property, slaves had no freedoms unless they were specifically granted to them.
There were several different types of slavery in the United States, depending on the geographic area. Most people think of "slavery" in terms of the plantation slaves, most of whom had little autonomy. They worked on the plantations picking tobacco and cotton, both of which required a lot of physical labor. On the other hand, in the lowlands of the Carolinas where rice was a common crop, slaves had slightly more autonomy, due to the low level of care required to grow rice. Slaves in this region often had other jobs, such as bookkeeping and smithing, and they were often sent to other farms to handle these tasks. However, they could not keep any money made from these jobs, as that went straight to their owners. In short, they had slightly more autonomy, but little freedom.