A transformer works based on electromagnetic induction - the changing magnetic flux induced by one coil induces a voltage in the other coil. Furthermore, any resistors attached to the coils of the transformers are still subject to Ohm's laws. Changes in the secondary coil also affect the primary coil. This is probably best illustrated with an example.
Let's take a 5V RMS AC source, connected to a step-up transformer with a 1:10 ratio. A 100 Ohm resistor is connected to the secondary coil. Let's try to work out the current being drawn.
The voltage across the secondary coil is pretty straight forward to work out:
Applying Ohm's law:
Then we can find the current across the primary:
Notice that if we were to attach a 50 Ohm instead of a 100 Ohm resistor, we would double the current through the secondary coil and also double the current through the primary coil. This is probably one of the concepts students find hardest to grasp - changing the resistance across the secondary coil changes the current in the primary coil as well as the secondary coil. Why? It's a bit difficult to explain, but it's to do with the primary coil inducing a back EMF on itself. For this reason, you also can't treat the primary coil as a conventional 'circuit' - the coil induces a back EMF which almost totally opposes the voltage drawn, even if there are no physical resistors in its path. Also, the current produced by a power source is not fixed - it can vary and will vary depending on the external circuit, such as what kind of transformers and resistors it is attached to.
So Ohm's law still applies in circuits with transformers, but we also have to take into account the effect of electromagnetic induction, and remember that generally speaking the current generated by a power source is not constant, but varies depending on the circuit it's attached to.