Powering up TEC plates is relatively easy, and the circuits depends mainly if you want some sort of regulation feedback, or if you just want to keep them powered with a stabilized current (they work in current, not in voltage), regardless to the effective temperatures.
If you want an extra-precise and extra-fast response feedback and temperature regulation, you can also buy a custom module or use an Arduino board, if instead you want just to experiment, or have a working, cheap circuit, here are some
First one, no feedback, is my version of the current sink LD driver .....
it's designed for laser diodes, but can be used for any load that need to be powered in current ..... simply don't mount the red components, that are planned for the LD modulation, connect the TEC plate in place of the LD, and use a power supply voltage a bit more high than the one required from the TEC plate (as example, if you have a 4,5V TEC plate, use 6V, if you have a 10V plate, use 12V, and so on,
but never go under the minimum working voltage of the op-amp):
Second, my old PWM driver based on the 555 IC and with a temperature feedback, place the NTC on the cold side of the assembly (preferably on the object, not directly on the TEC plate surface), and it simply shorten the distance from the pulses when the cold side temperature increase (increasing the medium value of the voltage), and elongate them when the temp decrease (decreasing the medium value), keeping the temperature relatively constant (not a true thermostat, but is the more cheap PWM alternative that i've designed til now

) ..... for change the frequency of the PWM pulses, if you want it more high, the 10uF capacitor connected from the GND and the pins 2 and 6 can be reduced, til 1uF, without too much problems:
And, if you want also to place some sort of control on the hot side, you can use one of these cheap solutions for change the speed of the fan placed on the cold side heatsink of the TEC (they increase the speed, so the airflow, when the temperature of the heatsink rise, proportionally ..... i made 2 versions, so regardless if you have N-channels or P-channels mosfets, you can use that what you can find more easily)
Just as suggestions
EDIT: i changed the 555 TEC driver a bit, for improve it. My version was working good, but another user had problem of instability, replicating it, so i made a pair of tests, and improved it, making it working more "smooth" and making the "on" and "off" parts of the PWM signal independently controlled ..... need just to add an 1N4148 (or any other silicon signal diode) between the pins 2-6 and 7, with the cathode connected to the pins 2-6 (is the one drawed in orange) ..... now, the trimmer control the "on" part of the signal (active pulses), and the NTC control the "off" part (pauses between the active pulses) independently, making the circuit working better also with critics and/or unstable versions of the 555 ..... also, changing the capacitor from 10uF to 1uF, make it work more stable 