So far, I've ordered the parts and tools that I think I need to be able to pull this off.
I've not soldered components as small as electronics before, so I also ordered a MintyBoost kit from Adafruit to learn and practice on. I've completed it and it wasn't as hard as I thought it might be.
I'm working my way through making sure that the parts for the base station are working and am making sure that I'm familiar with each part by itself. I've been able to make the temperature/humidity sensor (DHT22), the barometric pressure/temperature sensor (BMP180 breakout from Adafruit), and 20x4 LCD screen work on their own without much difficulty.
I'm currently in the process of working on getting the LCD to switch from displaying temperature, to displaying humidity, to turning off when a button is pressed. Apparently shutting off the LCD entirely isn't totally trivial, but I think I've found a way to do it with some extra components and some extra code.