Benefits of multiple op-amp gain stages in series?
In addition to improving the gain-bandwidth product of the circuit, splitting the amplifier into multiple stages allows you to use choose different op amps which are designed to excel at particular characteristics. For example, you can choose an op amp with good input characteristics (i.e. low offset, low noise, etc.) for the first stage and a (possibly different) op amp with good output characteristics (maximum output voltage swing, maximum output current, etc.) for the last stage. With only one stage you would have to find a jack-of-all-trades op amp that has good enough input and output characteristics (not to mention a high enough gain-bandwidth product).

The input characteristics of the first stage op amp are most important since all of that op amp's input non-idealities (offset, noise, etc.) are fully amplified along with the signal (since they are amplified by all stages). Non-idealities in the op amps for the second, third, etc. stages are not fully amplified and are not as much of a concern. In contrast, the first stage op amp doesn't need good output characteristics since its output will not swing as much as later stages and is driving a relatively high impedance load (the next op amp stage).

The last stage op amp can have the worst input characteristics since the signal at its input is nearly fully amplified and is much larger than the op amp's offset, noise, etc. The last stage op amp needs good output characteristics, though. For example, the op amp's maximum output voltage swing must be sufficient for the required signal output voltage swing (8 Vrms in your case), and it must have a sufficient slew rate for your amplified signal. The last stage op amp may also need to drive a low impedance load, in which case it needs to be able to source/sink more output current.

If noise is a concern, you can also consider using additional op amp active bandpass filter stages to reduce out-of-band noise. These stages may not provide signal gain, but they would improve the performance of the overall amplifier.



To give a concrete example, I once designed a low noise microphone pre-amplifier based on the TLE2027 low noise precision op amp. It has very good input characteristics, but its output characteristics aren't the best. In particular, its slew rate is only guaranteed to be on the order of \$1\text{ V}/{\mu\text{s}}\$ across temperature (the spec limit varies among versions -- see the datasheet). However, for an 8 Vrms output signal at 20 kHz you would need a slew rate of \$8\text{ V}\times \sqrt{2} \times 2\pi\times 20{\text{ kHz}} \approx 1.4\text{ V}/{\mu\text{s}}\$. It also isn't rail-to-rail at the output -- the output signal might be clipped using this op amp, depending on your supply voltages (e.g. if you used 9 V batteries). You would probably need to use a different op amp for the last stage in your amplifier.