How do AC adapters achieve low current?
Semi-ideal transformers assumed.


  So what is really going on? Do they make these coils with materials
  that have really high resistance or am I missing something?


You are over-complicating things by thinking about a transformer plus bridge and capacitor. Think about just the transformer with nothing connected to the secondary. Then, after thinking about it you should realize that if the secondary is not connected to anything, it can be removed. This leaves you with the primary connected to your AC mains circuit and that primary is now just an inductor.

So, if the inductor is (say) 10 henries in value, it will have an impedance at 50 Hz of \$2\pi f L\$ = 3142 ohms and will take a current of 73.2 mA when the applied AC voltage is 230 volts.

Additionally, because this current is created by the inductance of the primary, there is no power associated with it i.e. the current waveform is lagging the voltage waveform by 90 degrees: -



If it were a resistor of 3142 ohms the current would be in-phase with the voltage thus: -



Picture source.