Bringing Up Baby (or at least a new PCBA)
We have designed over 100 PCBAs, and each time I power one on for the first time I still get a little nervous. When we receive a new batch of boards, I go through the following process.
Visual Inspection - First, look over the board carefully. Take note of build quality, looking for crooked components and poor solder joints. Also as part of this inspection, try to gently rub some of the components off. If they are soldered on correctly then they will not flick off.
Power - I set the power supply to zero volts and connect it to the PCBA. Then I slowly increase the voltage while monitoring current, so if the current is too high (or too low) I can stop the test. Current too high indicates a short somewhere; current too low indicates a missing connection. If the board powers up without any issue then we measure power rails to ensure that the regulators are working correctly.
Start with the microcontroller - If the power test works correctly then the next step is to start programming the microcontroller, ensuring that JTAG works etc.
Work outward - Then work your way outward from there, adding peripherals, testing them as you go along. When you power up the board, keep an eye on current consumption - if it inexplicably spikes then you may have a problem.
This being said, if I'm working on a new circuit where I'm not sure if part of the circuit is correct, I'll just populate that part and leave the rest of the PCB unpopulated. For example, a recent project had a DAC feeding an audio output to a speaker; something that I haven't done much before. I populated the audio filter and PA and tested that before I put down anything else.