Puff of smoke.

I set up my Nano clone on a breadboard with a few things on the I2C bus (DS3231, 20x4 LCD, BMP180) and a AM2302 on pin 3.

When I first plugged it in a puff of smoke came out of the board (from underneath I think?) I immediately unplugged and pulled it out of the breadboard to look at it, a few minutes later I plugged it back in and uploaded the blink sketch.

I disconnected 5v and 3.3v rails on the breadboard and plugged it back in and it wouldn't turn on at all (no power light)

A few minutes later I plugged it in and it booted up into blink just fine, but now it wont program anything.

What are some good troubleshooting steps to see if I can salvage it or not?

Look at the board. Locate the scorch mark - where is it? Picture maybe?

My guess is you have damaged the USB serial chip, the regulator, the diode, or several of these.

You should not have plugged it back into the breadboard and powered it up until you had determined what happened.

I smoked the "auto-select" diode on the back of one once (it's a Schottky.)
Since it's black, it didn't stand out as burned; not till I put my other glasses on.
You can check it with a multimeter (set to diode function or low-ohms) - disconnected from power.