So just the UART socket would then serve the serial console?
Or would I have to connect to the JTAG socket?
No, JTAG is not necessary to access the serial console.
There are 4 pin's on the JTAG socket and 8 on the UART.
The other way around: 4 pins UART, 8 pins JTAG.
So far I was unable to find a cable/adapter that hooks into any of those ports. Could you point me to the right product there?
I'm not aware of any product that you could use out of the box (this is not to say they don't exist).
What I meant is that you could make one yourself. Buy the components, solder them, etc.
Or you could modify a product with a similar function.