There aren't any particular tests you should do. After a power surge it's difficult to determine any specific part of the machine that would have failed. Ideally, you'd have a surge protector in place that would have stopped any damage to the machine. If not, then hopefully you'd have a power supply with functionality built in. And if that's not the case, then it's literally anyone's guess.
What does make sense is to make sure all of your backups are up to date in case something does fail, as if another part has failed it's usually not too difficult to replace. Your data however may be irreplaceable, and as such, that should be top of your concerns.