The resistor is dissipating E^2/R=1.25*1.25/4.3=0.39W so a 1/2W resistor is more common and cheaper.
C1 has to be purchased (
look here).
When mounting the LM317 (TO220 package), it cannot short to the car frame. I would screw it to the car body using a nylon screw, with an insulating mica washer between it and the car metal. That way, no heatsink is required.
btw- reading some of the previous posts, only the heat produced by the regulator itself and the power dissipated in the resistor is conducted to the "heatsink". That power is ~1W in the regulator and ~0.4W in the resistor for a total of 1.4W.
OTOH, each LED is mounted someplace else, and it dissipates 3.3*0.3 or about 1W, so Carl is right, you will need to devise a mounting for each LED which mounts it mechanically, but conducts heat out of it to the car body.
The resistor can go either way around and be soldered directly between the Vout and Vadj pins.
The capacitor I show above is not polarized, and one lead should be soldered to Vin and the other grounded to the car body near the regulator.
The LEDs are polarized; one lead is called anode, and the other is cathode. The string of three LEDs is wired anode of 1 to Vadj;
cathode of 1 to anode of 2;
cathode of 2 to anode of 3;
cathode of 3 to car body. If you are mounting the three LEDs in different locations, you will have to daisy-chain a wire as needed.
Power feed wire to Vin. Connect other end to fused connection in the car's fuse block. 1A fuse is sufficient.