How many stages of transistor amplifiers should be used in order to produce an audible output
What is wrong with this circuit?


Typically a circuit like this has a gain that is roughly numerically the collector impedance divided by emitter impedance. So if you had 2k2 in the collector and 220 ohms in the emitter you'd get a voltage gain of ten.

This type of circuit is usually pushed to have much higher gains by bypassing the emitter resistor with a capacitor. This makes the emitter impedance very low at audio frequencies but also significantly increases distortion.

Then if you really want things to go wrong you make collector impedance something really low by hanging a 4 ohm speaker on it. Collector impedance (AC) falls from 2k2 to about 4 ohms and you have virtually no gain and lots of distortion.


  Reducing the value of emitter bypass capacitor reduces distortion but
  also reduces the gain. How do I find the correct value for getting
  good performance at audio frequencies?


Use a resistor in series with C1 to control the AC gain.


  Using this single stage I am able to get very less audio output. Its
  almost inaudible. If I use a second stage, I get highly distorted
  output. However it is audible to certain extent. What should I do to
  reduce the distortion without affecting the gain?


See my observations above - don't hang a speaker on the collector - use an emitter follower or better still get an audio amplifier IC like the LM386.


  Can you please provide the design steps involved in choosing the
  capacitor values?


I think this has been done.