I would use a 7809 regulator. 100uF and 0.1uF 25v input to ground. 0.1uF output to ground. The ground is the tab and 2nd pin.No need for anything more complicated. In the example forget the bridge rectifier as that will be your battery input.
Design of 9 V regulator using 7809 with circuit diagram and schematics
I would like to run a 9vdc powered preamp with a 12v car battery.