Usually I do a detailed visual inspection first, and if everything looks ok (no bent pins touching, no visible solder breaks, wires OK etc) then I power up one item at a time and watch for anything abnormal (heat, smoke, flame!, anything out of ordinary). Motors you can usually tell visually and by feeling them weather the bearings or case were damaged. The FC is usually a "try it and see" situation, after everything checks out I always assemble without props and do all the same run up tests you would do before a maiden, just paying extra attention to vibration, heat, smoke, abnormal attitude behavior (from motor RPM changes even if it's sitting still. I then strip them of anything extra like FPV gear and do a <10' altitude hover....and just go from there as I get comfortable with it again just as a maiden craft. The only real pieces of equipment you really should use if you can is an IR thermometer to check the ESC's and motors after you start doing real flights again and a simple multimeter with continuity for checking for any shorts. (this helps if you have a baseline of "pre crash" temps. Also if you crashed in anything soft like dirt/grass/gravel, take apart each motor and just check for debris stuck in the windings and cooling holes...especially if you sank one or two into the ground!
I'm sure there are people that whip out the scopes and metes and sensors etc and start NASA checking every connection/PCB component and that's totally fine too, just takes a lot longer, costs a lot more money (equipment) and IMO isn't necessary unless your dealing with a really high dollar copter or if your just a geek and into testing the crap out of things! I'm sure there's stuff I forgot, but that's the basics!
