Which transistor and resistor(s) do I need to hook up the FSA-410BT smoke alarm to Raspberry Pi's gpio?
From the datasheet/instructions you can see that the alarm provides for a remote LED and a floating NO/NC switch. The LED may be your best option.
You should use this to activate an optocoupler that will drive the Raspberry Pi I/O pin. 

I'd recommend you use something like the H11L1 which can work with your 3.3 V levels on the 'Pi.

 







simulate this circuit &ndash; Schematic created using CircuitLab

You also have a NO/NC floating contact you could use and the Alarm NO contacts (though this I'd recommend you don't use if you have other connections here).