OK. Might be fun!
A few points about the example circuit:
Since the TDA2030AV is designed to work with dual supply rails (positive and negative voltages on pins 5 and 3, respectively), to operate on a single rail, you have to shift your input signal to around 1/2 of the positive rail. That is why the two 100kΩ resistors on the input side are connected between the positive rail and ground. Except for some small AC signals, the voltage at that point between them will remain at nearly 1/2 the supply voltage at all times. The 22uF cap next to the lower one acts as a buffer to help keep it stable at that voltage.
The 2.2uF input capacitor blocks any DC portion of the input signal (your board should already have this cap, though it may not be the exact same value) but allows the AC audio signal to pass. The 100kΩ resistor placed after the input capacitor and between the non-inverting input of the op-amp and the center point between the other two 100kΩ resistors is there to tie that mid-voltage rail to the input of the op-amp. With this in place, the voltage at the op-amp input will wiggle around the mid-point of the supply rail.
The combination of the 100kΩ resistor and the 4.7kΩ resistor connected as shown between the output and the inverting input of the op-amp control the gain of the amp. For discussion’s sake, I’ll call the 100k resistor R1 and the 4.7k resistor R2. The formula for calculating the gain is:
So, the gain of this circuit is about 21 volts per volt, which can also be referred to as 27dB. As mentioned above, the gain of the LA4425A-E is given as 45dB, which translates to a gain of 178 volts per volt. To increase the gain of the TDA2030AV, you would increase the value of R1.
To get to the same 45dB level as the LA4425A-E, R1 would need to be approximately 831kΩ. The datasheet gives most of its specifications assuming a gain of around the 20-27dB range, so it is not clear what the effect of increasing the gain that much would be. I would recommend you use a 1MΩ multiturn trim pot for R1 and test your circuit at around 100kΩ first. Once it’s working, then you can start to increase the resistance incrementally and test as you go to make sure it still operates without excessive distortion.
The other components are there to properly filter the output and to protect the amp from oscillations and voltage spikes on the output.
Finally, most of the specifications given for the TDA2030AV assume a power supply of +/-16V, which equates to 32V between +Vs and ground on a single supply topology. The recommended minimum supply is +/-6V, which equates to a 12V single supply rail. It shouldn’t hurt it to go below this by a little bit, but it’s output power would be reduced, and if too low, it simply won’t operate.