Transistor & MOSFET driver circuits are "similar" but you can't just drop one in place of the other. And in either case, if the solenoid voltage is higher than the 5V processor/controller voltage you need an NPN transistor or an N-Channel MOSFET. I assume the solenoids are 12 or 24V.
A transistor is "easier" to turn-on because it takes less than 1V and MOSFETs are generally better for high-current. The 3906 is only rated for 200mA so apparently you don't need a lot of current.