Using F series TTL to drive a fet is probably not too good an idea. You'll find your top fets will probably stay turned on as 12V - 5V = 7V which normally will turn the fet on. You need a 'hi-side' driver for this or suitable discrete circuitry to drive the top (or hi side) fet.

I didn't check to see if the fets you choose are 'logic level' or standard threshold fets - this can be critical.

You rely on the parasitic diodes inside the fet - make sure they're rated to handle the reverse energy (probably not an issue but worth checking). If you're going to use a reverse emf diode - don't use a 1N4007 or suchlike - you need something faster if you expect to run the motor at a reasonable speed. Something like a UF4007.

You have no protection for the circuit - what happens if the motor shorts - you get fried fets. You need some sort of current limiting circuitry to make your system robust.

With stepper motors, the higher the voltage you hit them with, the faster they will step with adequate torque. Of course, the more voltage = more current. There are many techniques of doing this, from resistors, to pwm current control and microstepping. PWM is probably the best, but also the more complicated - the L297/293 chips take care of most of this. To build this out of discrete components takes some work. Obviously your supervisor wants you to learn a bit about power electronics. I doubt it is wise not to use any ic's - but I think his intention is for you not to use an integrated solution.