What kind of appliances what sort of actions do you want the Arduino to perform ?
One example would be to lower my window AC unit by 5 degrees, which would be done by 5 presses of a membrane button. Another example would be remotely controlling an old garage door (a single button press). I don't need to read any display information, just short the circuit for a moment as if it was my finger pushing the physical button instead. And I suspect that most of these are low-voltage circuits. (Let's assume I can access the solder points of the original appliance button.)
I imagine the answer to my question has something to do with wiring and coordinating the 5v arduino system with the unknown voltage of the appliance circuit and choosing the appropriately spec'd MOSFET.
Is this the simple relay functionality you were describing?
I'd love to hear there's a one-size fits all mosfet that would work in a variety of situations.