Energy from pickup coil
If I have an inductive coil generating AC voltage due to an
  oscillating magnetic field, is there a way to determine the current
  (or energy) being generated in the coil?


Yes this is quite simple, connect a load to the terminals using a fixed value resistor of a known value and measure the voltage across that resistor. Use ohms law etc..


  Further, if a capacitor were placed in parallel, would the energy get
  transferred to the capacitor and create a decaying oscillation?


Yes, an initial amount energy would be taken from the field to produce a constant oscillation (if the field is maintained). Energy cyclically interchanges between the coil's inductance and the capacitor and this is improved by choosing the capacitor value to match the coil inductance and operating frequency of the (presumably) AC magnetic field. There is a simple formula to calculate this.


  I know that the coil will generate a voltage to resist the changing
  magnetic field


No, the induced voltage does not resist the applied changing magnetic field.


  but does that mean a current flows through the coil equal to the
  voltage divided by coil resistance?


Without a load connected to the coil terminals, no current flows and if a load is connected then that current is determined by the self inductance of the coil, the incident magnetic field amplitude, the number of turns, the coil shape/dimensions, the operating frequency and the load impedance.