Generally speaking you get an immediate save, and you get a save each round because it says " The subject can attempt a Fortitude saving throw each round to reduce the damage by half".
That first round still counts as one of the rounds.
I want to clarify exactly when a character gets a saving throw against a spell and the effect of that saving throw. This is for spells that don't have "Saving Throw negates" in the spell description. So for a spell such as Trial of Fire and Acid that specifies when the target gets a save, assuming the spellcasting is successful (all concentration checks made, etc.), does the target get an immediate saving throw to avoid the entire effects of the spell plus a save each round OR is the target affected without making any saving throw, and then makes the save per round to halve the damage for just that round?
In general, you get one immediate saving throw, and otherwise are subject to the effects of the spell as described. However, many spell descriptions offer additional opportunities to save.
The initial save is what is given in the "Saving Throw" line in the spell's description (see CRB p. 216). In this case, the initial save is against half of the initial damage.
If the target suceeds on the initial save, it's still under the effect of the spell and will take damage every round.
Basically, this happens: Round one: Presuming the touch attack is successful, the target makes a Fortitude saving throw to prevent half of 2d6.
Round two: No matter the result of the saviong throw made last round, the target makes a Fortitude saving throw to prevent half of 2d6.
Round three til whenever the duration runs out: Same as in round two.
Note that there is a check to avoid the entire spell, but the caster makes the roll and not the target (the touch attack).