Unwanted noise/oscillations on op amp output
As already mentioned, some bias on the input would be an idea if you are only using a single supply - half the supply voltage supplied through resistors with a parallel value of > 10x the piezo impedance. You also need the bias voltage at the base of the Rf/Rg resistors then. Due to the high impedances involved, you will have to be careful with layout, wire lengths, etc. 

As for the offset, you can setup an adjustment pot that injects some current into the negative input to counteract this (and/or get a really low offset opamp - you can get FET trimmed opamps with as low as 100uV offset)

Something like this (some parts left out such as caps, etc) I set the opamp up with 6mV offset, you can see the difference in the simulation:



Sim without adjustment:



Sim with adjustment: