It probably doesn't actually need a relay as a thyristor should do it
Thyristors have a very different switching behaviour compared to a relais. Once triggered by a gate pulse, they stay on until the load current drops (close) to zero. On a DC supply, a thyristor stays on until power is switched off or the motor burns out. However, with pulsing DC current (i.e. AC after a rectifier, but no capacitor), the thyristor will switch off at the end of a half wave. This allows tricks like a crude PWM regulator by triggering the thyristor somewhat after the begin of a half wave (Proxxon uses this trick for their "DC" minimot tools).
For driving AC motors, a thyristor won't help you much, as it either isolates or behaves like a diode, so the AC motor gets either no current or pulsed DC.
For full control of a DC motor on a DC supply, you typically use an H-bridge setup with bipolar or MOSFET transistors. This allows PWM for speed control, turning left and right, braking (not with every controller) and coasting. There are severel very clever H-bridge driver ICs available that add a lot of protection for both motor (e.g. stall and overcurrent detection) and driver transistors (overcurrent, short circuit detection). Some H-bridge driver ICs even include the transistors. And yes, there are cheap, ready-to-use modules with H-bridges.
H-Bridge drivers either accept a direction and a speed signal (on/off or PWM), or a left-turn and a right-turn signal (again either on/off or PWM). If breaking is supported, the direction-and-speed signals need a third signal to switch between breaking and coasting, the left/right approach can use the illegal state (left and right on at the same time) for breaking, or also use a third signal line. After all, driving a solid-state H-bridge is not too different from driving two relays. Two "slow" bits are sufficient for switching left/off/right, you may need to add a third bit for breaking. With a direction-and-speed driver, change one "slow" bit to PWM output to allow speed control.
L298-based modules seem to be very popular (two motors up to 46 V, each 3 A abs max peak, each 2 A operating, logic supply 4.5 V to 7 V, logic levels should be just about compatible with 3.3 V signals, motor drivers can be paralleled for double output current), just add a heatsink, four diodes per motor and some passives to the chip and you have a driver module. Only downside is that the L298 is everything but smart, no protection except for shorting out the motor supply.
Today I will gladly share my knowledge and experience, for there are no sweeter words than "I told you so". ;-)