Thermostat is to the right of the alternator, and left of the intake manifold. Just look for where the lower radiator hose runs to a housing on the front of the engine, that housing will have a sensor and associated wiring attached to the end.
Need a bit more information on the car and smoke on startup. How many miles do you have on this 2006? Following and keeping up with normal maintenance? What color is the smoke? How much is there - how long does it continue? Is there a noticeable odor to it - oil, gasoline, sweet, acrid, etc.? Anything else seems to be out of the ordinary - check engine lights, oil consumption, poor fuel economy, rough idle, poor driveability, no heat from heater, windows fogging up, etc.?
Might be related to why you are wanting to replace the thermostat? Do you know the thermostat is truely faulty, or is this part of your diagnostic process?
Like Dave mentioned, you can take this discussion to Corolland - the sister site of Toyoland that deals mainly with Corollas. Or we can continue with this topic here.