To drive the motor in both directions, you need an H-bridge. See attached diagram. Note that if you make the mistake of driving both output pins high at the same time, you will short out the 12v supply, so a fuse is highly recommended.
The built-in diode in a power mosfet is not a problem, it is an advantage because in a half-bridge or full-bridge configuration, it acts as a flyback diode.
Alternatively, use an H-bridge chip such as the L293D (if you are certain that the motor stall current is less than about 0.5A) or the L298 (for higher current), but expect to lose up to 3.6v or so in the chip, so you will need to increase your 12v supply to 15v or so.