Ah, geez, so... Fixed it this afternoon.
Remember, I said I'm new to electronics...
The Arduino and LEDs were running off separate DC power supplies - a 9v one for the Arduino and a selectable one for the LEDs (since I intended to up the voltage as needed when I added more LEDs).
I started poking around with a multimeter and noticed the voltage readings were not what they should be.
The emitter leg of the PN2222 was grounded to the DC power supply that was powering the LEDs. The 5v coming from the Arduino output pin that was triggering the base pin of the PN2222 was apparently not liking that ground, and I was getting negative voltage readings between those two points.
Since I'm only running in the 6-7.5v range for now, I powered the Arduino off the selectable power supply as well and all was working as expected.