How to repurpose an old PCB?
I am brand new to electronics and would like to know if there's any way to reprogram the microcontroller or reuse a used PCB.

Not universally. Most devices that contain a microcontroller will have programmed the microcontroller to disable any further programming. Or, the microcontroller was 1-time-programmable to begin with.
I don't even think there's a programmable microcontroller on your board, at all! There's simply application-specific controllers that integrate Bluetooth, SD card reader, and audio logic, and do everything a company without any own knowledge of these rather hard fields needs to produce a cheap-as-possible Bluetooth speaker.
Look at that PCB: It's really quite cheaply made, and doesn't need many components to do what is a rather complicated task!
Reusing such a (not really great-looking) PCB also doesn't pay: Never quite what you need, usually. If you want to get started into microcontroller programming:
All microcontroller companies make &quot;evaluation boards&quot;, to help people get started using their microcontrollers.
And the ones from ST for their STM32, called Nucleo, are cheap, so start with one of these that contains the peripherals you want.

Also, I don't recognize which part is used as the Bluetooth or FM module or antenna on the PCB.

The zig-zag line on the bottom right of your picture is a very crude 2.4 GHz antenna, so that's your bluetooth. The closest IC to that is the Bluetooth IC.
FM antennas (you mean FM in the broadcast band ca 88–108 MHz) are far too large to fit on the PCB, so whatever does the FM uses the audio or power supply cabling as antenna. That works well because the audio frequencies are much, much smaller than the radio frequencies.

My question might be really basic or silly but I don't know much and am trying to learn, so please help.

Not silly, just reflecting you've not got much experience. That's normal!