I think that you should consider a detailed functional study of the product at the operator level first. Then you can understand the operational behaviors of the components of the system.
As far as buttons and selectors: These are so easy to understand in terms of function that there is little to "reverse engineer" with them. Simple momentary push switches can easily be understood to connect to your MCU via GPIOs and pull-up resistors. You may be faced with debouncing these inputs in software. You will just have to reproduce the input detection on these inputs as actions into the replicated software functionality that you will be coding. Similar ideas apply to rotary selector wheel type devices.
From a display standpoint you should be able to quickly determine what type device that it is.....a character mode device or a graphics device. A logic analyzer could prove quite useful to determining the initialization and operational protocols for the display unit. If it is a display module you may get lucky and find a part number on it or in its controller chip. From this you can likely find a data sheet that describes the display controller protocols rather than having to reverse engineer them.