Converting to low voltage thermostat

So I recently purchased a condo in a cold weather climate. Everywhere I've ever lived has had central heat/air. My new place has central heat and air conditioning, but it also has some baseboard heaters in the living room. I'm from the South and have never had a heater like this, but it has a knob with a water stain beneath it. I give this info because it might be important for what I want to do (or hopefully won't be at all).

This place was built in the 60s and has what is likely the original thermostat still installed: a Honeywell T651A thermostat. Here is is new, and here is what mine looks like now without the cover. It works perfectly (as far as I can tell). It cools, and it heats. However I want to get with the times and install a programmable, low-voltage thermostat system. Options for new line voltage thermostats don't accomplish what I want.

So, my set up is this thermostat on the wall, AC intake and motor/fan in the ceiling above it, and wired to my breaker box which is in another room. There is only this thermostat in the condo.
Question is, what is the process for converting to a low voltage system? I'm literate and can read/figure out wiring diagrams/do basic wiring, but I want to understand what parts I'll need and whether it is difficult so I know if I can do this myself. What would it take, and could I make to where something like the Nest thermostat could be used?