but I can't figure out what the second stage is or how it works? In
which stage does the amplification take place?
Sure you can figure it out, you just need a little help.
If you recall that the base-emitter voltage of a transistor operating in the active region is nearly constant, then you can figure out that the 2nd transistor cannot be a voltage amplifier; the signal voltage on the emitter is almost the same as the signal voltage on the base.
So, the voltage amplification must be due to the 1st transistor circuit. This transistor is configured as a classic common emitter amplifier.
The reason for the 2nd transistor circuit may not be immediately obvious but it is, in fact, crucial to the proper operation of this amplifier.
The speaker is a very low impedance load. For significant voltage gain, the collector of the 1st transistor must be connected to relatively high impedance as the gain is proportional to this impedance.
If you connect the speaker (through the coupling capacitor) directly to the collector of the 1st transistor, the speaker's impedance is in parallel with the collector resistor so the collector is now connected to a very low impedance and thus, the voltage gain falls to nearly zero.
However, the 2nd transistor is configured as a common collector amplifier which acts as a voltage buffer. Essentially, looking into the base of the 2nd transistor, the 8 ohm speaker impedance is multiplied by the beta (plus 1) of the 2nd transistor.
If the beta is 100, the speaker impedance "looks" 101 times larger through the base so, by connecting the base of the 2nd transistor to the collector of the 1st transistor, some voltage gain is still possible from the 1st stage.