Is there a way to identify what exactly I fried?
Sometimes the chip will get hot when powered-up.
With the microcontroller, sometimes you can check to see if the clock is running (with an oscilloscope). If it's not, the chip is probably fried. (It's possible to have a partially-bad microcontroller, but usually the whole chip dies.)
If you have an UNO board with the plug-in chip it's certainly worth trying another one. Worst case, it doesn't help and you've got an extra chip. Make sure to buy an "Arduino" chip with the bootloader pre-loaded! A "blank" Atmega chip is useless unless you have a way to program it.
Of course you can check the voltages. If 5V and/or 3.3V are low, it could be a bad/blown regulator chip, or it could be a shorted chip "dragging down" the voltage.
If you have a stand-alone programmer you can try programming the chip. The bootloader isn't much help because there are several things that can make that fail, but if the bootloader works, the chip is OK.
If you over-voltage or reverse-voltage a board, of course there is a possibility of burning-up more than one chip.
The thing about troubleshooting/debugging is... You NEVER know for-sure what the problem is (was) 'till it's fixed. And, SOMETIMES after it's fixed you don't know what the problem was!
Is the Arduino made to be idiot proof, or am I just getting lucky?
I had a shorted output pin once, and it was shorted for at least several minutes, or maybe an hour... Long enough for me to troubleshoot it, and panic when I discovered the short! But, it survived with no problems.
I had another Arduino die "for no reason" (probably static discharge). I usually ground myself before touching electronics but at home I don't have a ground strap. It an SMT version and I didn't attempt to repair it.
I've never over-voltage'd an Arduino or ATmega chip. But at work, we have a 5V board that I've over-voltage'd a few times.... If I apply 12V the RAM chip will die every time. Sometimes the CPU survives and sometimes not. Sometimes another chip will burn-up too. But, my adjustable "5V" bench power supply only goes to about 7V and I've never hurt a board at 7V, so I'm safe as long as I'm using that power supply.