From a 'cold' start, if your temp rises quickly it usually means a cracked head from my experience and of course, the tell tale white smoke is usually a blown head gasket (coolant getting into the combustion process)
If I were you I'd drain the coolant, access the thermostat, remove (hopefully without damaging rubber gasket) look around the base of the plunger on the T-stat or spring for scuffing or rubbing marks proving it's cocking as it moves. If so, replace. If not drill the little holes that I recomended a few threads up and re-instsall with one or more holes in the 'up' position, again just using water, run with bleeder craked open (looks like a 12mm bolt with a little tower on top with a hole thru the center. Like a BIG brake bleeder.
Have that open enough to get a good flow of water/steam/bubbles coming out. Slowly tighten said bleeder until when you rev the motor using the accelerator cable to about 2000rpms and NO more foam/bubbles come out. Just coolant or water.
If you get a lot of foam/bubbles, tighten bleeder, hold upper rad hose ( in addition to keeping an eye on the gauge) and feel for expansion or the compression from the combustion process getting imto the cooling system. If so, something is cracked or degraded somewhere.
On a average motor, the temp should take several min to even get warm notwithstanding the fans coming on. Some of my cars fans wouldnt' come on for about 15+min. But since it gets ssoo hot down here that I just can't rely on a thermo switch going bad, So I just bypass them where the fan is switched on with the ign key.
I hope this can help you or others.
E
