To protect a circuit from shorting and for polarity protection for the battery should I use a transistor or a fuse?
A fuse doesn't usually protect your circuit from anything. It protects the stuff around your circuit from catching fire after your circuit has already blown itself up.
If you want to protect the circuit itself from damage, you should use some kind of transistor or diode circuit. Exactly what circuit you should use depends on what kind of scenarios you want to protect from (ESD, over-voltage, reverse voltage, load dump, ...), and how much energy might be available from the mis-connected power source.