Darn you V=IR, you are wrong (does V really equal IR?)
The only reason the current would be zero is if you have a non-zero inductance (which all circuits do). Once you factor in the inductance, you'll be able to calculate the rise time of the current after you apply a voltage. Only in the ideal world where there's no inductance or capacitance, will V=IR be true at all times over a resistor.

In our non-ideal world, you're right that V=IR only applies at steady state after the influence from capacitance and inductance fall away.