I use an H bridge with 2 Pchannel and 2 Nchannel FETs on a 13.5 volt rail. The TEC pulls about 8 amps when flat out. The power supply can deliver 15 amps and the FETs can switch 75 amps. The H bridge is controlled via a dedicated 16F88 which reads a Microchip MCP9800 to give me 1/16 degree resolution and drives the H bridge. Because the FETs have just simple resistive gate drive via 2N7002 as level shifters there MUST be delays inserted between turning the FETs On or Off otherwise there is a short but deadly overlap where both FETs on the same side of the H bridge are conducting simultaneously. This leads to the FETs getting very hot or worse, it burns the traces off the PCB. My application is fully proportional and must both heat and cool. The drink cooler examples I have seen are simple cool-only with a thermostat to drop the drive once below the setpoint. Drive is flat out or off.

You MUST ensure the entire surfaces of each side of the TEC are thermally bonded to the cool chamber or the heatsink otherwise any small unbonded area will get very hot and burnout the TEC in no time.
TECs are poor coolers but good heaters. My device works between 1 Celsius and 45 Celsius -roughly Ambient +/- 25 Celsius. In cooling mode I must blow lots and lots of air over the heatsink to get down to the lower temperatures and supply high currents. When heating the chamber the average drive currents are about 1/4 of the cooling currents.
TEC life is severely degraded by thermal stress. If your controller drives full heat then full cool so that the average gives you the setpoint you are after the life will be short. The thermal stresses will break the bonds between the magic metal elements. I run a fixed frequency 50 Hz software PWM and vary the duty cycle from 0 up to about 19500 microseconds. I am only running a Proportional controller, not a full PID. Seems to regulate to about 0.1 Celsius but is slow getting to the setpoint which does not matter in my application which is left on for days on end.

HTH
BrianT