What are some steps to attach an Arduino to an electric heater?
Would or should begin? Don't fiddle with the internals of the heater. 

Set its switch to the always on position and control switching on and off with an external relay. Make a box that has plenty of room (you can reduce size at a later moment), a power cord, a mains adapter for your Arduino and a mains outlet in which you can plug your heater.

Use:


an RTC (real time clock) breakout to keep accurate time
optionally a 7 segment display to display date/time. Just make sure it has a driver chip on it so you don't need your Arduino to multiplex the individual LEDs
couple push buttons to configure the device
a relay rated for the maximum power of the heater to turn it on/off, that can be driven from 5V. Possibly needs a transistor and flyback diode to be controlled from an Arduino GPIO line.


And be very careful when experimenting with mains power, it can be leathal.