Choosing resistor values for Op-amp and inverting/non-inverting configuration
As far as I'm aware the ADC on an arduino requires 0V to 5V as the range of input and because you have a signal that is -1V to +1V, you need to centre it about 2.5V. If the signal is purely AC then that is easily done with a potential divider and capacitor: -



Note C1, R1 and R2. C1 blocks the average 0V level of the input signal and R1 and R2 set the input side to the op-amp at half-Vdd. You probably don't need the op-amp - just feed the junction of R1 and r2 into the arduino. This is just a convenient circuit I found that is useful for explaining C1, R1 and R2 action.