All final hardware decisions are listed and Amazon-linked in the components section.
I started with my goal: to remotely check if the garage door is closed for peace-of-mind.
I decided that I didn't want to interfere in any way with the existing opener unit because I'd rather not break it trying to interface with it and I don't want to control it because that introduces a security risk. I decided to use a non-contact sensor to reduce wear on the door and/or sensor as well as any scratching noise or resistance to movement. This left one good option: an ultrasonic distance sensor. I can mount it near the bottom of the door opening or on the ceiling and check if it detects a door close in front of it.
I have an Arduino UNO, however, WiFi shields are expensive and I'd rather keep it available for tinkering. While researching WiFi-enabled microcontrollers, I stumbled across this article about the ESP8266. I found a nicer one for just under $10 on Amazon and now it's on its way.