How can I detect my vehicle's shifter position? Will a Hall Effect Sensor work?
This is actually harder than it looks to get repeatable.  

I would not recommend hall effect sensors for this application.  Their sense distances are generally small, meaning that the mounting of the sensors and magnet need to be done with some precision.  You will find it very challenging to align all of the halls to work consistently and under all operating conditions.

The transmission will move on its compliant mounts as a function of engine torque.  If you're dealing with a top-loader transmission, the shift lever will also move relative to the body as a function of engine torque, making it almost impossible to obtain repeatable measurements.

Edit:  The OP has a top-loader style transmission, where there is no external linkage to take advantage of.  For any other transmission, there is external linkage between the shifter and transmission housing - usually rods or cables.  

For a top loader:  I'd recommend X and Y axis potentiometers on the shifter, with the pots mounted to the transmission (not the body), which should be accessible through the floorpan opening that the shifter comes up through.  You'll need to decode the analog readings from the two pots to match gear selection.  

For any other transmission:  I'd recommend using a sealed potentiometer used as a voltage divider connected to each shift linkage (under the body, not inside the transmission).  You'd then do the calibration + windowing in your firmware, where it will be easier to tune.  If your transmission has multiple linkages, it essentially de-muxes some of the multidimensionality out of the shifter for you, again making it easier to map discrete gears to analog voltage ranges.

Reverse is easy--just tap the back up light circuit.