Opamp for amplifying 1V signal voltage to 2V I'm doing my first electronics project where a signal that varies between -1 to 1 V is going into an analog pin on the Arduino. The amplified signal has to be read as fast as possible by Arduino. The source signal is at 60 Hz.

Will this LM324 be a suitable 4-channel opamp for my needs? How concerned should I be about the 2 mV input offset and 20 nA input current bias?

To make full use of the 10bit ADC, would you recommend using an opamp with gain of 2 and biasing the signal by +2.5V? If so, will it be better to create the bias using a potential divider across +5V and GND, or using a 2.5V voltage regulator?

Or would it be better to set AREF to 2V and bias the signal by +1V? In this case, will it be better to use a 2V voltage regulator, or a voltage divider across Arduino's 3V3 and GND using 20M and 33M resistors?
A 2mV offset voltage will become a 4mV error after amplification. You haven't shown the schematic for your amplification, but the current bias will also become error after amplification.

That LM324 variant is surface mount, but there are other LM324 variants in a DIP package, which will work fine in a breadboard.