Some of your inputs and the LCD can potentially be multiplexed, but not enough to fit within the number of I/O pins on a Uno or Leonardo. So I suggest you use an Arduino Mega - that will be much simpler than messing around with I/O expanders or shift registers. You can find 8-channel relay boards on eBay and elsewhere.
btw the modern way to set temperatures etc. is to use a rotary encoder or up/down buttons while displaying the set point on the LCD, not potentiometers.