Most efficient way to convert multiple voltages?
The choice between using a simple 3 terminal linear supply and a switch mode power supply depends on the current you need, as well as the difference between the input voltages you have and the output voltages you need.

You did not specify the current draw requirements for the LED strip and the Arduino. You specify that you expect to use a buck converter for the LED strip, why is this? If the input voltage is only slightly higher than the regulated voltage, a high current low-dropout linear regulator will be almost as efficient, for much less complexity. Recall that the formula for efficiency in a linear regulator is Efficiency ~= Vout/Vin; if Vout is very close to Vin, then the efficiency will be close to 100%. A low-dropout linear regulator such as the LT1185 regulating from 15V to 12V will be approximately 80% efficient, which is better than many switch-mode regulators.

Your real question is how to power the Arduino, which runs at 5V. In this case, it is true that the efficiency of regulating from 12V down to 5V via a linear regulator will be poor, approximately 40%. However, your total power loss is proportional to current: P_loss = I_load*(Vin-Vload). The Arduino's exact current draw depends on which peripherals are enabled and what the code is doing, but assuming you are not doing much more than using the serial port to display some patterns, a current draw of 50mA is reasonable. Therefore, your loss in using a common linear regulator (such as a 7805) is only 350mW, which is reasonable if your application is not battery powered.