I have noticed that a MOSFET symbol has a little diode in it (or at least what looks like a diode). Does this mean I do not have to worry about using a diode in a circuit that runs a motor using one? I would have used a diode in order to prevent reverse voltage from turning the motor by hand.
Yes, power MOSFETs have a parasitic diode called Body Diode.
As a result of this diode, a single MOSFET can work only as a unidirectional switch. A single MOSFET can't switch-off the opposite direction, because the diode conducts independent of the gate.
The body diode is usually fairly slow to turn on. I advise against using it as a the only flyback diode. Add an external Schottky diode for flyback protection.
(There are power MOSFETs with a built-in Schottky diode. They have 2 diodes: the native body diode, and an additional Schottky. But such MOSFETs are relatively uncommon.)